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Palindromic Sums of Powers
rood Wonplate 178



Introduction

Palindromic numbers are numbers which read the same from

 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 7, 3113, 44611644



Palindromic Sums of Powers

The powers expressionThe palindromic curioLength
B.S.Rangaswamy [ June 15, 2009 ] curio 9
605 + 533 + 17027777777779
B.S.Rangaswamy [ July 11, 2009 ] curio 10 - minimum one 3rd, 4th or higher power
230 + 61132 + 172 + 152 + 22111111111110
231 + 86452 + 502 + 72222222222210
577352 + 502 + 242 + 25333333333310
16443 + 10702 + 113 + 152 + 22444444444410
129 + 198942 + 153 + 242 + 24555555555510
18823 + 913 + 1102 + 33666666666610
881912 + 503 + 142 + 102777777777710
233 + 172902 + 104 + 142888888888810
999992 + 4472 + 53 + 26999999999910
B.S.Rangaswamy [ July 11, 2009 ] curio 20
33333333332 + 326 + 61132 + 172 + 152 + 221111111111111111111120
47140452072 + 926422 + 203 + 54 + 222 + 1022222222222222222222220
57735026912 + 1017302 + 463 + 402 + 243333333333333333333320
66666666662 + 942802 + 553 + 632 + 1224444444444444444444420
74535599242 + 1220512 + 4722 + 152 + 1325555555555555555555520
81649658092 + 672872 + 105 + 402 + 636666666666666666666620
88191710362 + 1247252 + 513 + 782 + 1127777777777777777777720
94280904152 + 1243942 + 533 + 202 + 53 + 528888888888888888888820
99999999992 + 1414212 + 105 + 262 + 34
OR
46415883 + 73402262 + 11222 + 182 + 35
9999999999999999999920



Dt. 17th July 2009
Dear Mr Patrick

                                    SUB: More Curios

                     Please find herewith enclosed summation of powers with common base numbers, culminating in palindromes as below:

                     ? 2^41 + 2^34 + - - - - - - - - - -  =  2222222222222 ( 13 Dgts )

                     ? 3^30 + 3^27 + - - - - - - - - - -  =  222222222222222 ( 15 Dgts )
                     
                     ? 5^21 + 5^19 + - - - - - - - - - -  =  555555555555555 ( 15 Dgts ) 

                     ? 7^14 + 7^12 + - - - - - - - - - -  =  777777777777 ( 12 Dgts )


                     ? 2^25 + 2^10 + - - - - - - - - - -  =  33555533 ( 8 Dgts )

                     Enclosed is an Excel sheet depicting the above features, for your kind perusal and considerations.

                     It is felt that palindromes with much larger number of digits can be formulated on similar lines.


With Regards

B.S.Rangaswamy

Note: Sheet1 may be viewed for reference.

1 2 3 5 7 11 2 4 9 25 49 121 3 8 27 125 343 1331 4 16 81 625 2401 14641 5 32 243 3125 16807 161051 6 64 729 15625 117649 1771561 7 128 2187 78125 823543 19487171 8 256 6561 390625 5764801 214358881 9 512 19683 1953125 40353607 2357947691 10 1024 59049 9765625 282475249 25937424601 11 2048 177147 48828125 1977326743 285311670611 12 4096 531441 244140625 13841287201 3138428376721 13 8192 1594323 1220703125 96889010407 34522712143931 14 16384 4782969 6103515625 678223072849 379749833583241 15 32768 14348907 30517578125 4747561509943 4177248169415650 1 16 65536 43046721 152587890625 33232930569601 45949729863572200 17 131072 129140163 762939453125 232630513987207 505447028499294000 18 262144 387420489 3814697265625 1628413597910450 5559917313492230000 19 524288 1162261467 19073486328125 11398895185373100 61159090448414500000 20 1048576 3486784401 95367431640625 79792266297612000 672749994932560000000 21 2097152 10460353203 476837158203125 558545864083284000 7400249944258160000000 22 4194304 31381059609 2384185791015620 3909821048582990000 81402749386839800000000 23 8388608 94143178827 11920928955078100 27368747340080900000 895430243255237000000000 24 16777216 282429536481 59604644775390600 191581231380566000000 25 33554432 847288609443 298023223876953000 1341068619663960000000 26 67108864 2541865828329 1490116119384770000 9387480337647750000000 27 134217728 7625597484987 7450580596923830000 65712362363534300000000 28 268435456 22876792454961 37252902984619100000 459986536544740000000000 29 536870912 68630377364883 186264514923096000000 3219905755813180000000000 30 1073741824 205891132094649 931322574615478000000 22539340290692300000000000 31 2147483648 617673396283947 32 4294967296 1853020188851840 33 8589934592 5559060566555520 34 17179869184 16677181699666600 35 34359738368 50031545098999700 36 68719476736 150094635296999000 37 137438953472 450283905890997000 38 274877906944 1350851717672990000 39 549755813888 4052555153018980000 40 1099511627776 12157665459056900000 41 2199023255552 36472996377170800000 42 4398046511104 109418989131512000000 43 8796093022208 328256967394537000000 44 17592186044416 984770902183611000000 45 35184372088832 2954312706550830000000 46 70368744177664 8862938119652500000000 47 140737488355328 26588814358957500000000 48 281474976710656 79766443076872500000000 49 562949953421312 239299329230618000000000 50 1125899906842620 4 717897987691853000000000 51 2251799813685250 8 52 4503599627370500 496 53 9007199254740990 2 54 18014398509482000 1984 55 36028797018964000 3968 56 72057594037927900 36 57 144115188075856000 5872 58 288230376151712000 1744 59 576460752303423000 488 60 1152921504606850000 46976 61 2305843009213690000 3952 62 4611686018427390000 87904 63 64 65
A S q u a r e s S N Power Value 1 2^41 2199023255552 2 2^34 17179869184 3 2^32 4294967296 4 2^30 1073741824 5 2^29 536870912 6 2^26 67108864 7 2^25 33554432 8 2^23 8388608 9 2^22 4194304 10 2^18 262144 11 2^13 8192 12 2^9 512 13 2^8 256 14 2^7 128 15 2^3 8 16 2^2 4 17 2^1 2 * 2222222222222 13 Digit Palindrome 1 2^25 33554432 2 2^10 1024 3 2^6 64 4 2^3 8 5 2^2 4 6 2^0 1 * 33555533 8 Digit Palindrome OR 1 2^25 33554432 2 10^3 1000 3 10^2 100 4 10^0 1 * 33555533 8 Digit Palindrome
B C u b e s S N Power Value 1 3^30 205891132094649 2 3^27 7625597484987 3 3^27 7625597484987 4 3^25 847288609443 5 3^23 94143178827 6 3^23 94143178827 7 3^22 31381059609 8 3^21 10460353203 9 3^19 1162261467 10 3^19 1162261467 11 3^17 129140163 12 3^15 14348907 13 3^14 4782969 14 3^14 4782969 15 3^12 531441 16 3^12 531441 17 3^10 59049 18 3^10 59049 19 3^8 6561 20 3^8 6561 21 3^7 2187 22 3^7 2187 23 3^6 729 24 3^5 243 25 3^5 243 26 3^3 27 27 3^3 27 28 3^1 3 222222222222222 15 Digit palindrome * Power < 2
C 5th Powers S N Power Value 1 5^21 476837158203125 2 5^19 19073486328125 3 5^19 19073486328125 4 5^19 19073486328125 5 5^19 19073486328125 6 5^17 762939453125 7 5^17 762939453125 8 5^17 762939453125 9 5^15 30517578125 10 5^15 30517578125 11 5^15 30517578125 12 5^15 30517578125 13 5^14 6103515625 14 5^14 6103515625 15 5^13 1220703125 16 5^11 48828125 17 5^11 48828125 18 5^10 9765625 19 5^10 9765625 20 5^10 9765625 21 5^9 1953125 22 5^9 1953125 23 5^9 1953125 24 5^9 1953125 25 5^8 390625 26 5^8 390625 27 5^7 78125 28 5^5 3125 29 5^5 3125 30 5^4 625 31 5^4 625 32 5^4 625 33 5^3 125 34 5^3 125 35 5^3 125 36 5^3 125 37 5^2 25 38 5^2 25 39 5^1 5 555555555555555 15 Digit palindrome
D 7th Powers S N Power Value 1 7^14 678223072849 2 7^12 96889010407 3 7^11 1977326743 4 7^10 282475249 5 7^10 282475249 6 7^9 40353607 7 7^9 40353607 8 7^9 40353607 9 7^7 823543 10 7^7 823543 11 7^6 117649 12 7^6 117649 13 7^6 117649 14 7^6 117649 15 7^6 117649 16 7^6 117649 17 7^4 2401 18 7^3 343 19 7^3 343 20 7^3 343 21 7^2 49 777777777777 12 Digit Palindrome
Dt. 25th July 2009 Dear Mr Patrick SUB: Few More Curios In continuation of my submissions dated 17th July, please find herewith enclosed details of Repdigital Palindromes together with their constituents for 11th,13th, 17th & 19th powers for your kind perusal and considerations. With Regards B.S.Rangaswamy
Dt.25.07.2009 Dear Mr Patrick SUB: Few more Curios E 4444444444444444 (16 Digits) - 13 = 11^15 + 7(11^13) + 8(11^12) + 11^11 + 5(11^10) + 6(11^9) + 3(11^8) + 3(11^7) + 8(11^6) + 5(11^5) + 8(11^3) + 9(11^2) + 10(11^1) F 222222222222 (12 Digits) - 9 = 13^10 + 7(13^9) + 12(13^8) + 5 (13^7) + 6(13^6) + 2(13^5) + 9(13^3) + 13^2 + 11(13^1) G 9999999999999999 (16 Digits) - 11 = 17^13 + 2(17^11) + 13(17^10) + 5(17^9) + 11(17^8) + 17^7 + 7(17^6) + !7^5 + 4(17^4) + 10(17^2) + 13(17^1) H 888888888888888888 (18 Digits) - 14 = 19^14 + 2(19^13) + 2(19^12) + 11(19^11) + 11(19^10) + 2(19^9) + 7(19^8) + 8(19^7) + 7(19^6) + 10(19^5) + 9(19^4) + 11(19^3) + 2(19^2) + 7(19^1) Following are revised details of Repdigital palindromes, submitted earlier: A 2222222222222 (13 Digits) - 17 = 2^41 + 2^34 + 2^32 + 2^30 + 2^29 + 2^26 + 2^25 + 2^23 + 2^22 + 2^18 + 2^13 + 2^9 + 2^8 + 2^7 + 2^3 + 2^2 + 2^1 B 222222222222222 (15 Digits) - 18 = 3^30 + 2(3^27) + 3^25 + 2(3^23) + 3^22 + 3^21 + 2(3^19) + 3^17 + 3^15 +2(3^14) + 2(3^12) + 2(3^10) + 2(3^8) + 2(3^7) + 3^6 + 2(3^5) + 2(3^3) + 3^1 C 555555555555555 (15 Digits) - 16 = 5^21 + 4(5^19) + 3(5^17) + 4(5^15) + 3(5^14) + 5^13 + 2(5^11) + 2(5^10) + 4(5^9) + 2(5^8) + 5^7 + 2(5^5) + 2(5^4) + 4(5^3) + 2(5^2) + 5^1 D 777777777777 (12 Digits) - 10 = 7^14 + 7^12 + 7^11 + 2(7^10) + 3(7^9) + 2(7^7) + 6(7^6) + 7^4 + 3(7^3) + 7^2 Please see Enclosed Excel sheets 2 & 1 for details. Excel tables are now revised for Cubes, 5th & 7th powers. With Regards B.S.Rangaswamy
11 13 17 19 121 169 289 361 1331 2197 4913 6859 11 14641 28561 83521 130321 9 161051 371293 1419857 2476099 10 1771561 4826809 24137569 47045881 7 19487171 62748517 410338673 893871739 8 214358881 815730721 6975757441 16983563041 7 2357947691 10604499373 118587876497 322687697779 2 25937424601 137858491849 2015993900449 6131066257801 11 285311670611 1792160394037 34271896307633 116490258898219 11 3138428376721 23298085122481 582622237229761 2213314919066160 1 2 34522712143931 302875106592253 9904578032905940 42052983462257100 0 59 2 379749833583241 799006685782884000 121 4177248169415650 45949729863572200 505447028499294000 5559917313492230000 61159090448414500000 672749994932560000000 7400249944258160000000 81402749386839800000000 895430243255237000000000
A S q u a r e s S N Power Value 1 2^41 2199023255552 2 2^34 17179869184 3 2^32 4294967296 4 2^30 1073741824 5 2^29 536870912 6 2^26 67108864 7 2^25 33554432 8 2^23 8388608 9 2^22 4194304 10 2^18 262144 11 2^13 8192 12 2^9 512 13 2^8 256 14 2^7 128 15 2^3 8 16 2^2 4 17 2^1 2 * 2222222222222 13 Digit Palindrome 1 2^25 33554432 2 2^10 1024 3 2^6 64 4 2^3 8 5 2^2 4 6 2^0 1 * 33555533 8 Digit Palindrome OR 1 2^25 33554432 2 10^3 1000 3 10^2 100 4 10^0 1 33555533 * 8 Digit Palindrome
B C u b e s S N Power Value 1 3^30 205891132094649 2 2(3^27) 15251194969974 3 3^25 847288609443 4 2(3^23) 188286357654 5 3^22 31381059609 6 3^21 10460353203 7 2(3^19) 2324522934 8 3^17 129140163 9 3^15 14348907 10 2(3^14) 9565938 11 2(3^12) 1062882 12 2(3^10) 118098 13 2(3^8) 13122 14 2(3^7) 4374 15 3^6 729 16 2(3^5) 486 17 2(3^3) 54 18 3^1 3 * 222222222222222 15 Digit palindrome * Power < 2
C 5th Powers S N Power Value 1 5^21 476837158203125 2 4(5^19) 76293945312500 3 3(5^17) 2288818359375 4 4(5^15) 122070312500 5 2(5^14) 12207031250 6 5^13 1220703125 7 2(5^11) 97656250 8 2(5^10) 29296875 9 4(5^9) 7812500 10 2(5^8) 781250 11 5^7 78125 12 2(5^5) 6250 13 3(5^4) 1875 14 4(5^3) 500 15 2(5^2) 50 16 5^1 5 * 555555555555555 15 Digit palindrome
D 7th Powers S N Power Value 1 7^14 678223072849 2 7^12 96889010407 3 7^11 1977326743 4 2(7^10) 564950498 5 3(7^9) 121060821 6 2(7^7) 1647086 7 6(7^6) 705894 8 7^4 2401 9 3(7^3) 1029 10 7^2 49 777777777777 12 Digit Palindrome
Dt. 29.07.2009 Dear Mr Patrick Please find herewith enclosed 'Curio 32' for your kind perusal. Thank you immensely for calling me to participate in this highly interesting and educating Topic. I have used more than one square or cube or other powers in each of my findings, which may not match your ultimate requirement. It is for others to explore and arrive at such high value palindromes. With Regards B.S.Rangaswamy
C U R I O - 32 (32 Digit palindromes) 11111111111111111111111111111111 (7,7) = 3333333333333333^2 + 47140452^2 +2730^2 + 17^3 + 3^4 + 2^4 + 2^3 OR = 165^14 + 158437088969584^2 + 16236309^2 + 5051^2 + 18^3 + 2^9 + 2^2 22222222222222222222222222222222 (6,7) = 4714045207910316^2 + 88425577^2 + 7430^2 + 120^2 + 11^2 + 2^4* OR = 8286^8 + 37887508351616^2 + 8195697^2 + 221^3 + 38^3 + 52^ 2+ 2^6 33333333333333333333333333333333 (7,7) = 5773502691896257^2 + 86306864^2 + 10450^2 + 79^2 + 3^3 + 2^4 + 2^2 OR = 734^11 + 130542669028054^2 + 6563007^2 + 217^3 + 45^3 + 85^2+ 11^2 44444444444444444444444444444444 (7,7) = 6666666666666666^2 + 94280904^2 + 5461^2 + 19^3 + 46^2 +12^2 + 3^5 OR = 9036^8 + 28078249904977^2 + 5208833^2 + 101^3 + 13^3 + 2^6 + 2^3 55555555555555555555555555555555 (6,6) = 7453559924999298^2 + 121361815^2 + 10722^2 +85^2 + 3^2 + 2^3 OR = 7453559924999298^2 + 121361815^2 + 486^3 + 421^2 +5^2 + 2^2 66666666666666666666666666666666 (6,7) = 8164965809277260^2 + 73110759^2 + 5657^2 + 6^4 + 2^5 + 2^3 OR = 8164965809277260^2 + 73110759^2 + 317^3 + 52^3 + 19^3 +17^2 + 6^3 77777777777777777777777777777777 (6,7) = 8819171036881968^2 + 105832141^2 + 526^3 + 222^2 + 2^3 + 2^2 OR = 33^21 + 607082836486390^2 + 19150661^2 + 261^3 + 17^2 + 7^2+ 2^2 88888888888888888888888888888888 (6,7) = 9428090415820633^2 + 111445726^2 + 14720^2 + 150^2 + 14^2 + 3^3 OR = 9428090415820633^2 + 111445726^2 + 14720^2 + 28^3 + 2^9 + 3^5+ 2^4 99999999999999999999999999999999 (7,6,6) = 9999999999999999^2 + 141421356^2 + 406^3 + 58^3 + 13^3 + 2^9 + 5^2 OR = 9999999999999999^2 + 109^8 + 8624012^2 + 349^2 + 10^2 + 2^5 OR = 46415888336^3 + 28739148791^2 +141768^2 + 442^2 + 7^2 + 5^2 Following is a random palindrome, dictated by my grandson over telephone: 37985621462109866890126412658973 (5,6) Its constituents are 6163247639200445^2 + 90681241^2 + 7427^2 + 23^2 + 3^2 OR 33615513156^3 + 47496382543^2 + 271934^2 + 715^2 + 10^2 + 3^3 OR - - - - - - - - - - - - - - - - - - - - - - - With the liberty to use Squares, Cubes and other powers more than once, Every palindrome can be expressed as summation of powers in several ways!! To arrive at the least number of constituent powers is really is an intellectual task. - B.S.Rangaswamy
> From: patrick.degeest@skynet.be > To: psdevices@hotmail.com > Subject: RE: C U R I O 32 > Date: Mon, 3 Aug 2009 01:27:56 +0200 > > Dear B.S.Rangaswamy, > > Thanks a lot for the latest submissions regarding the repdigits > expressed in a variety of sums of powers. They are indeed very > nice but alas are not directly solutions as I intended them > for wonplate 178. The reason is that I think they are > 'engineered' rather than 'sporadic'. In that spirit the > sumpower of your grandson is more like it and I will add > his palindrome to the plate. > > Of course I wouldn't like to lose all those nice repdigital > palindromes that you send in. Instead I will create a dedicated > but separate page for these numbers, in due time. Please be > patient and all will come to a good end. As soon as it is finished > I'll ask you to proofread it and then it can get uploaded. > > Best regards, > > P@rick. > > Patrick De Geest > 1. mailto:pdg@www.worldofnumbers.com > 2. mailto:Patrick.DeGeest@skynet.be > website 1 : http://www.worldofnumbers.com/index.html > website 2 : http://users.skynet.be/worldofnumbers/ (mirrorsite) > > > > Van: B.S.Rangaswamy . [mailto:psdevices@hotmail.com] > > Verzonden: woensdag 29 juli 2009 10:07 > > Aan: patrick DeGeest > > CC: psdevices@hotmail.com > > Onderwerp: C U R I O 32 > > > > Dear Mr Patrick > > > > Kindly see enclosures > > > > Your sincerely > > > > B.S.Rangaswamy > > > > Dear Mr Patrick In thanking for your earnest feed back on my findings and inclusion of 32 digit palindrome of my grandson together with its constituents,you may take appropriate action on my other findings in this regard . With Regards B.S.Rangaswamy
Dear Mr Patrick Further to my earlier reply, I definately agree that all the findings submitted by me are engineered, and not spariadic as desired/pronounced. Regret for diversifying on this very interesting subject topic. Your Sincerely B.S.R
Dt. 6th August 2009 Dear Mr Patrick Kindly excuse me for disturbing and drawing your attention to a few of my remaining findings on some more repdigit palindromes generated from powers of base numbers 23, 29, 31, 37, 41, 43, 53, 73 and 79 vide enclosures for your kind perusal. It is true that these are not sporadic, but engineered for varied variety of exploration. I only wish we can come across some very enchanting palindromes with many scores of digits as sum of powers, through your renowned website World! of Numbers. With Regards B.S.Rangaswamy
Repdigit Palindromes as Sum of powers of primes B.S.Rangaswamy 1111111111111111111111 (22 Digits) - 14 Constituents = 4(23^15) + 3(23^14) + 19(23^13) + 9(23^12) + 18(23^11) +14(23^10) + 4(23^9) + 9(23^7) + 8(23^6) + 23^5 + 16(23^4) + 8(23^3) + 2(23^2) + 6(23^1). 1111111111111111111111111111 (28 Digits) - 18 Constt. = 5(29^18) + 8(29^17) + 3(29^16) + 29^15 + 27(29^14) + 10(29^13) + 5(29^12) + 27(29^11) + 10(29^10) + 24(29^9) + 17(29^8) + 27(29^7) + 24(29^6) + 18(29^5) + 28(29^4) + 11(29^3) + 2(29^2) + 24(29^1). 333333333333333 (15 Digits) - 9 Constt. = 12(31^9) + 18(31^8) + 25(31^7) + 20(31^6) + 9(31^5) + 6(31^4) + 3(31^3) + 12(31^2) +18(31^1). 666666666666666 (15 Digits) – 9 Constt. = 5(37^9) + 4(37^8) + 29(37^7) + 21(37^6) + 16(37^5) + 19(37^4) + 37^3 + 15(37^2) + 16(37^1). 333333333333333 (15 Digits) – 8 Constt. = 41^9 + 30(41^7) + 22(41^6) + 36(41^5) + 34(41^4) + 22(41^3) + 8(41^2) + 20(41^1). 333333333333333333333 (21 Digits) – 12 Constt. = 8(43^12) + 14(43^11) + 29(43^10) + 38(43^9) + 29(43^8) + 15(43^7) + 9(43^6) + 29(43^5) + 15(43^4) + 34(43^3) + 25(43^2) + 4(43^1). 9999999999999 (13 Digits) – 7 Constt. = 8(53^7) + 27(53^6) + 9(53^5) + 13(53^4) + 45(53^3) + 34(53^2) + 3(53^1). 3333333333333333 (16 Digits) – 8 Constt. = 4(73^8) + 9(73^7) + 53(73^6) + 22(73^5) + 2(73^4) + 73^3 + 62(73^2) + 12(73^1) 2222222222222 (13 Digits) – 6 Constt. = 9(79^6) + 11(79^5) + 15(79^4) + 6(79^3) + 46(79^2) + 71(79^1). - - - - - - - - - - - - - - - Note: No repdigit palindromes of 32 digits and below, can be generated from powers of primes 47, 59. 61, 67, 71, 83, 89 & 97 unless K(N^0) is used at the end, in each of these cases similar to generation of 8 digit palindrome 33555533 under powers of Two. Excel sheets for generating repdigital palindromes for powers of 2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 53, 73 and 79 are furnished at sheet 2 of enclosure - P O W R. - - - - - - - - - - - - - - - - - - - -
59 73 79 3481 5329 6241 205379 389017 493039 12117361 28398241 38950081 714924299 2073071593 3077056399 42180533641 151334226289 243087455521 2488651484819 11047398519097 19203908986159 146830437604321 806460091894081 1517108809906560 1 33 8662995818654940 58871586708267900 13
Sum of Powers Sum of Powers Sum of Powers Sum of Powers A Powers of TWO B Powers of THREE C Powers of FIVE D Powers of SEVEN E Powers of 11 F Powers of 13 G Powers 0f 17 H Powers of 19 I Powers of 31 J Powers of 37 K Powers of 41 L Powers of 53 M Powers of 73 N Powers of 79 S N Power Value S N Power Value S N Power Value S N Power Value S N Power Value + S N Power Value S N Power Value + S N Power Value + S N Power Value S N Power Value S N Power Value S N Power Value S N Power Value + S N Power Value 1 2^41 2199023255552 1 3^30 205891132094649 1 5^21 476837158203125 1 7^14 678223072849 1 11^15 4177248169415650 1 1 13^10 137858491849 1 17^13 9904578032905940 7 1 19^14 799006685782884000 121 1 12(31^9) 317275465928052 1 5(37^9) 649808698975385 1 41^9 327381934393961 1 8(53^7) 9397689118696 1 4(73^8) 3225840367576320 4 1 9(79^6) 2187787099689 2 2^34 17179869184 2 2(3^27) 15251194969974 2 4(5^19) 76293945312500 2 7^12 96889010407 2 7(11^13) 241658985007510 7 2 7(13^9) 74231495611 2 2(17^11) 68543792615260 6 2 2(19^13) 84105966924514000 118 2 18(31^8) 15352038673938 2 4(37^8) 14049917815684 2 30(41^7) 5842628216430 2 27(53^6) 598437750483 2 9(73^7) 99426586671870 3 2 11(79^5) 33847620389 3 2^32 4294967296 3 3^25 847288609443 3 3(5^17) 2288818359375 3 7^11 1977326743 3 8(11^12) 25107427013760 8 3 12(13^8) 9788768652 3 13(17^10) 26207920705830 7 3 2(19^12) 4426629838132000 322 3 25(31^7) 687815352775 3 29(37^7) 2753024436857 3 22(41^6) 104502293302 3 9(53^5) 3763759437 3 53(73^6) 8020713993310 7 3 15(79^4) 584251215 4 2^30 1073741824 4 2(3^23) 188286357654 4 4(5^15) 122070312500 4 2(7^10) 564950498 4 11^11 285311670610 1 4 5(13^7) 313742585 4 5(17^9) 592939382480 5 4 11(19^11) 1281392847880000 409 4 20(31^6) 17750073620 4 21(37^6) 53880254589 4 36(41^5) 4170823236 4 13(53^4) 102576253 4 22(73^5) 45607575040 6 4 6(79^3) 2958234 5 2^29 536870912 5 3^22 31381059609 5 2(5^14) 12207031250 5 3(7^9) 121060821 5 5(11^10) 129687123000 5 5 6(13^6) 28960854 5 11(17^8) 76733331850 1 5 11(19^10) 67441728835000 811 5 9(31^5) 257662359 5 16(37^5) 1109503312 5 34(41^4) 96075874 5 45(53^3) 6699465 5 2(73^4) 56796480 2 5 46(79^2) 287086 6 2^26 67108864 6 3^21 10460353203 6 5^13 1220703125 6 2(7^7) 1647086 6 6(11^9) 14147686140 6 6 2(13^5) 742586 6 17^7 410338670 3 6 2(19^9) 645375395000 558 6 6(31^4) 5541126 6 19(37^4) 35609059 6 22(41^3) 1516262 6 34(53^2) 95506 6 73^3 389010 7 6 71(79^1) 5609 * 7 2^25 33554432 7 2(3^19) 2324522934 7 2(5^11) 97656250 7 6(7^6) 705894 7 3(11^8) 643076640 3 7 9(13^3) 19773 7 7(17^6). 168962890 3 7 7(19^8) 118884941000 287 7 3(31^3) 89373 7 37^3 50653 7 8(41^2) 13448 7 3(53^1) 159 * 7 62(73^2) 330390 8 2222222222222 8 2^23 8388608 8 3^17 129140163 8 2(5^10) 29296875 8 7^4 2401 8 3(11^7) 58461510 3 8 13^2 169 8 17^5 1419850 7 8 8(19^7) 7150973000 912 8 12(31^2) 11532 8 15(37^2) 20535 8 20(41^1) 820 * 9999999999999 8 12(73^1) 870 6 * 13 Digit palindrome 9 2^22 4194304 9 3^15 14348907 9 4(5^9) 7812500 9 3(7^3) 1029 9 8(11^6) 14172480 8 9 11(13^1) 143 * 9 4(17^4) 334080 4 9 7(19^6) 329321000 167 9 18(31^1) 558 * 9 16(37^1) 592 * 333333333333333 13 Digit Palindrome 3333333333333290 43 10 2^18 262144 10 2(3^14) 9565938 10 2(5^8) 781250 10 7^2 49 10 5(11^5) 805250 5 222222222222 10 10(17^2) 2890 0 10 10(19^5) 24760000 990 333333333333333 666666666666666 15 Digit Palindrome 16 Digit palindrome 11 2^13 8192 11 2(3^12) 1062882 11 5^7 78125 777777777777 11 8(11^3) 10640 8 12 Digit Palindrome 11 13(17^1) 210 6 * 11 9(19^4) 1172000 889 15 Digit Palindrome 15 Digit Palindrome 12 2^9 512 12 2(3^10) 118098 12 2(5^5) 6250 12 Digit Palindrome 12 9(11^2) 1080 9 9999999999999950 49 12 11(19^3) 75000 449 13 2^8 256 13 2(3^8) 13122 13 3(5^4) 1875 13 10(11^1) 110 0 * 16 Digit Palindrome 13 2(19^2) 0 722 14 2^7 128 14 2(3^7) 4374 14 4(5^3) 500 4444444444444380 64 14 7(19^1) 0 133 * 15 2^3 8 15 3^6 729 15 2(5^2) 50 16 Digit Palindrome 888888888888882000 6888 16 2^2 4 16 2(3^5) 486 16 5^1 5 * 18 Digit Palindrome 17 2^1 2 * 17 2(3^3) 54 555555555555555 2222222222222 18 3^1 3 * 15 Digit palindrome 13 Digit Palindrome 222222222222222 15 Digit palindrome 1 2^25 33554432 * Power < 2 2 2^10 1024 3 2^6 64 4 2^3 8 5 2^2 4 6 2^0 1 * 33555533 8 Digit Palindrome OR 1 2^25 33554432 2 10^3 1000 3 10^2 100 4 10^0 1 * 33555533 8 Digit Palindrome
Repdigital palindromes as sum of multi powers of 101 The second lowest Prime Palindrome ‘101’ is an exceptional number, in the Numerical World. Its multiple powers can add up to innumerable Number of Repdigit palindromes (RDP) of digits, which are multiples of ‘4’. All RDPs of 8.12,16 digits together with their multiple powers of 101 are enclosed at Excel sheet 2 of POW 101. The fifteen constituents of RDP 11111111111111111111111111111111 (32 Digits) are: 9(101^15) 57(101^14) 63(101^13) 17(101^12) 95(101^11) 69(101^10) 33(101^9) 48(101^8) 39(101^7) 43(101^6) 32 (101^5) 89(101^4) 39(101^3) 81(101^2) 88(101^1). RDPs of 32 digits starting from 2, 3, 4, 5, 6, 7, 8 and 9 can all be expressed as sum of multiple powers of 101 in similar ways. RDPs of 36, 40, 44 - - - - digits can also be expressed as sum of multiple powers of 101. This phenomenon has no limits and can go on & on to infinite limits of the number of digits of RDPs in the World of Numbers!! - B.S.Rangaswamy Dt. 8th August 2009
4 3 107213535210701 62 6647239183063460 2 2 1061520150601 18 19107362710810 8 1 10510100501 30 315303015030 0 104060401 46 4786778440 6 1030301 30 30909030 0 5 10201 18 183610 8 4 101 62 6260 2 3 6666666666666640 26 2 1 107213535210701 72 7719374535170470 2 1061520150601 55 58383608283050 5 10510100501 1 10510100500 1 5 104060401 87 9053254880 7 4 1030301 68 70060460 8 3 10201 89 907880 9 2 101 5 500 5 1 7777777777777740 37 107213535210701 82 8791509887277480 2 5 1061520150601 91 96598333704690 1 4 10510100501 74 777747437070 4 3 104060401 28 2913691220 8 2 1030301 6 6181800 6 1 10201 58 591650 8 101 49 4940 9 8888888888888850 38 5 4 107213535210701 93 9970858774595190 3 3 1061520150601 27 28661044066220 7 2 10510100501 45 472954522540 5 1 104060401 69 7180167660 9 1030301 45 46363540 5 10201 27 275420 7 5 101 93 9390 3 4 9999999999999960 39
8 Digit Palindromes 12 Digit Palindromes 16 Digit Palindromes 16 Digit Palindromes S N Power Value S N Power Value S N Power Value + S N Power Value + 1 10(101^3) 10303010 1 10(101^5) 105101005010 1 10(101^7) 1072135352107010 0 1 62(101^7) 6647239183063460 2 2 79(101^2) 805879 2 57(101^4) 5931442857 2 36(101^6) 38214725421630 6 2 18(101^6) 19107362710810 8 3 22(101^1) 2222 3 76(101^3) 78302876 3 72(101^5) 756727236070 2 3 30(101^5) 315303015030 0 Total 11111111 4 35(101^2) 357035 4 41(101^4) 4266476440 1 4 46(101^4) 4786778440 6 5 33(101^1) 3333 5 38(101^3) 39151430 8 5 30(101^3) 30909030 0 1 21(101^3) 21636321 Total 111111111111 6 70(101^2) 714070 0 6 18(101^2) 183610 8 2 57(101^2) 581457 7 44(101^1) 4440 4 7 62(101^1) 6260 2 3 44(101^1) 4444 1 21(101^5) 220712110521 Total 1111111111111090 21 Total 6666666666666640 26 Total 22222222 2 14(101^4) 1456845614 3 51(101^3) 52545351 1 20(101^7) 2144270704214020 0 1 72(101^7) 7719374535170470 2 1 32(101^3) 32969632 4 70(101^2) 714070 2 73(101^6) 77490970993870 3 2 55(101^6) 58383608283050 5 2 35(101^2) 357035 5 66(101^1) 6666 3 43(101^5) 451934321540 3 3 101^5 10510100500 1 3 66(101^1) 6666 Total 222222222222 4 82(101^4) 8532952880 2 4 87(101^4) 9053254880 7 Total 33333333 5 77(101^3) 79333170 7 5 68(101^3) 70060460 8 1 31(101^5) 325813115531 6 39(101^2) 397830 9 6 89(101^2) 907880 9 1 43(101^3) 44302943 2 72(101^4) 7492348872 7 101* 8880 8 7 5(101^1) 500 5 2 13(101^2) 132613 3 27(101^3) 27818127 *101 2222222222222190 32 Total 7777777777777740 37 3 88(101^1) 8888 4 4(101^2) 40804 * Total 44444444 5 99(101^1) 9999 1 *101 3323619591531730 1 1 82(101^7) 8791509887277480 2 Total 333333333333 2 9(101^6) 9553681355400 9 2 91(101^6) 96598333704690 1 1 53(101^3) 54605953 3 15(101^5) 157851507510 5 3 74(101^5) 777747437070 4 2 93(101^2) 948693 1 42(101^5) 441424221042 4 23(101^4) 2393389220 3 4 28(101^4) 2913691220 8 3 9(101^1) 909 2 29(101^4) 3017751629 5 15(101^3) 15454510 5 5 6(101^3) 6181800 6 Total 55555555 3 2(101^3) 2060602 6 9(101^2) 91800 9 6 58(101^2) 591650 8 4 40(101^2) 408040 7 31(101^1) 3130 1 7 49(101^1) 4940 9 1 64(101^3) 65939264 5 31(101^1) 3131 Total 3333333533333300 33 Total 8888888888888850 38 2 71(101^2) 724271 Total 444444444444 3 31(101^1) 3131 1 41(101^7) 4395754943638740 1 1 93(101^7) 9970858774595190 3 Total 66666666 1 52(101^5) 546525226052 2 45(101^6) 47768406777040 5 2 27(101^6) 28661044066220 7 2 86(101^4) 8949194486 3 87(101^5) 914378743580 7 3 45(101^5) 472954522540 5 1 75(101^3) 77272575 3 78(101^3) 80363478 4 64(101^4) 6659865660 4 4 69(101^4) 7180167660 9 2 49(101^2) 499849 4 75(101^2) 765075 5 53(101^3) 54605950 3 5 45(101^3) 46363540 5 3 53(101^1) 5353 5 64(101^1) 6464 6 79(101^2) 805870 9 6 27(101^2) 275420 7 Total 77777777 Total 555555555555 7 75(101^1) 7570 5 7 93(101^1) 9390 3 Total 4444444444444410 34 Total 9999999999999960 39 1 86(101^3) 88605886 1 63(101^5) 662136331563 2 27(101^2) 275427 2 43(101^4) 4474597243 1 51(101^7) 5467890295745750 1 3 75(101^1) 7575 3 54(101^3) 55636254 2 82(101^6) 87044652349280 2 Total 88888888 4 9(101^2) 91809 3 59(101^5) 620095929550 9 5 97(101^1) 9797 4 4(101^4) 416241600 4 1 97(101^3 99939197 Total 666666666666 5 92(101^3) 94787690 2 2 5(101^2) 51005 6 49(101^2) 499840 9 3 97(101^1) 9797 1 74(101^5) 777747437074 7 18(101^1) 1810 8 Total 99999999 2 29(101^3) 29878729 Total 5555555555555520 35 3 45(101^2) 459045 4 29(101^1) 2929 Total 777777777777 1 84(101^5) 882848442084 2 58(101^4) 6035503258 3 4(101^3) 4121204 4 80(101^2) 816080 5 62(101^1) 6262 Total 888888888888 1 95(101^5) 998459547595 2 14(101^4) 1456845614 3 81(101^3) 83454381 4 14(101^2) 142814 5 95(101^1) 9595 Total 999999999999
10th August 2009 Dear Mr Patrick Please find enclosed revised page 1 0 1 - MPR. This is in place of 1 0 1 - MP, submitted earlier, on 6th August. The only revisions are: A. Serial numbers are added to the list of multipowers of 101. B. "infinite range" replaces the earlier "infinite limits" in the last paragraph. Thanking You Your sincerely B.S.Rangaswamy
Dt. 15th August 2009 Dear Mr Patrick Please find enclosed corrected multi powers of 101 for generating RDP of 32 digits starting with 1, vide 1 0 1 MPR for your kind perusal. Extremely sorry for the error committed earlier. This is now prepared in Excel sheet. All other 32 digit RDPs starting with 2, 3, 4 - - - -9 are also being prepared on Excel. I shall be submitting these to you on completion. Thanking you , With Regards B.S.Rangaswamy
Repdigital palindromes as sum of multi powers of 101 The second lowest Prime Palindrome ‘101’ is an exceptional number, in the Numerical World. Its multiple powers can add up to innumerable Number of Repdigit palindromes (RDP) of digits, which are multiples of ‘4’. All RDPs of 8.12,16 digits together with their multiple powers of 101 are enclosed at Excel sheet 2 of POW 101. The fifteen constituents of RDP 11111111111111111111111111111111 (32 Digits) are: 1. 9(101^15) 2. 57(101^14) 3. 63(101^13) 4. 17(101^12) 5. 95(101^11) 6. 69(101^10) 7. 33(101^9) 8. 48(101^8) 9. 39(101^7) 10. 43(101^6) 11. 32(101^5) 12. 89(101^4) 13. 38(101^3)* 14. 91(101^2)* * Corrected powers 15. 88(101^1) RDPs of 32 digits starting from 2, 3, 4, 5, 6, 7, 8 and 9 can all be expressed as sum of multiple powers of 101 in similar ways. RDPs of 36, 40, 44 - - - - digits can also be expressed as sum of multiple powers of 101. This phenomenon has no limits and can go on & on to infinite range of the number of digits of RDPs in the World of Numbers!! - B.S.Rangaswamy Dt.15th August 2009
Dt. 26th August 2009 Dear Mr Patrick Kindly excuse me for this belated submission. In continuation of my previous mails, please find herewith enclosed RPDT P, an Excel format of Sum of multiple powers of ‘101’ generating Repdigital palindromes of 32 digits starting with 1, 2, .. . .to.9, for your kind perusal. These are not sporadic, but engineered to derive repdigital palindromes. For your kind information sheet 1 of this format is ‘protected’ where as sheet 2 is unprotected. You can make use of sheet 2 for any likely additions / deletions. Your encouraging statements terming ‘Nine sevens’ as Gem; appreciating my patience at certain stage, have resulted in arriving patiently at these results. Every Repdigital palindrome, whose number of digits is a multiple of 4 can be expressed as sum of multiple powers of “101” the second lowest palindrome prime in the World of Numbers. This phenomenon can extend even up to infinite range! The random palindrome of 32 digits is expressed in three different ways, which is illustrated in sheet 3 of Excel. At C this palindrome is generated from multiple powers of the lowest palindrome prime “11”. I am highly grateful to you for bestowing me this opportunity to participate in this very interesting and educative topic, through your esteemed website. Thanking You, With Regards B.S.Rangaswamy
S U M O F M U L T I P L E P O W E R S O F " 1 0 1 " SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 9(101^15) = 104487205983 299868165026 O4463509 1 38(101^15) = 441168203040 599443363443 29957038 1 66(101^15) = 766239510544 199033210190 99399066 2 57(101^14) = 6552003015 454447178664 99949857 2 28(101^14) = 3218527797 O65342473730 17519228 2 100(101^14) = 11494742132 376223120464 91140100 3 63(101^13) = 71699876 667297233325 67221963 3 50(101^13) = 56904664 O21664470893 39065050 3 38(101^13) = 43247544 656464997878 97689438 4 17(101^12) = 191560 255122434852 51240417 4 71(101^12) = 800045 771393698501 66945271 4 24(101^12) = 270438 O07231672732 95868824 5 95(101^11) = 10598 849293320507 27354595 5 79(101^11) = 8813 779938656000 78536979 5 63(101^11) = 7028 710583991494 29719363 6 69(101^10) = 76 218926653373 11119069 6 75(101^10) = 82 846659405840 33825075 6 81(101^10) = 89 474392158307 56531081 7 33(101^9) = 360916139985 83909733 7 32(101^9) = 349979287258 99548832 7 32(101^9) = 349979287258 99548832 8 48(101^8) = 5197712187 O1478448 8 92(101^8) = 9962281691 77833692 8 35(101^8) = 3789998469 69828035 9 39(101^7) = 41813278 73217339 9 56(101^7) = 60039579 71799256 9 74(101^7) = 79338016 O5591874 10 43(101^6) = 456453 66475843 10 72(101^6) = 764294 50843272 10 28(101^5) = 2942 82814028 11 32(101^5) = 3363 23216032 11 30(101^5) = 3153 O3015030 11 19(101^4) = 19 77147619 12 89(101^4) = 92 61375689 12 54(101^4) = 56 19261654 12 70(101^3) = 72121070 13 38(101^3) = 39151438 13 54(101^3) = 55636254 13 37(101^2) = 377437 14 91(101^2) = 928291 14 64(101^2) = 652864 14 10(101^1) = 1010 15 88(101^1) = 8888 15 49(101^1) = 4949 777777777775 2770546105037 872185903 111111111108 3111111111105 605169154 444444444441 3357437499815 641429414 777777777777 777777777777 77777777 111111111111 111111111111 11111111 444444444444 444444444444 44444444 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 19(101^15) = 220584101520 299721681721 64978519 1 47(101^15) = 545655409023 899311528469 34420547 1 76(101^15) = 882336406081 198886726886 59914076 2 14(101^14) = 1609263898 532671236865 O8759614 2 86(101^14) = 9885478233 843551883599 82380486 2 57(101^14) = 6552003015 454447178664 99949857 3 25(101^13) = 28452332 O10832235446 69532525 3 12(101^13) = 13657119 365199473014 41375612 3 42(101^12) = 473266 512655427282 67770442 4 35(101^12) = 394388 760546189402 23142035 4 89(101^12) = 1002874 276817453051 38846889 4 58(101^11) = 6470 876410658836 O1963858 5 90(101^11) = 10041 O15119987848 99599090 5 74(101^11) = 8255 945765323342 50781474 5 49(101^10) = 54 126484145149 O2099049 6 37(101^10) = 40 871018640214 56687037 6 43(101^10) = 47 498751392681 79393043 6 65(101^9) = 710895427244 83458565 7 66(101^9) = 721832279971 67819466 7 66(101^9) = 721832279971 67819466 7 84(101^8) = 9095996327 27587284 8 96(101^8) = 10395424374 O2956896 8 39(101^8) = 4223141151 94951239 8 12(101^7) = 12865624 22528412 9 78(101^7) = 83626557 46434678 9 96(101^7) = 102924993 80227296 9 43(101^6) = 456453 66475843 10 86(101^6) = 912907 32951686 10 14(101^6) = 148612 82108414 10 61(101^5) = 6411 16130561 11 65(101^5) = 6831 56532565 11 63(101^5) = 6621 36331563 11 8(101^4) = 8 32483208 12 77(101^4) = 80 12650877 12 42(101^4) = 43 70536842 12 8(101^3) = 8242408 13 77(101^3) = 79333177 13 93(101^3) = 95817993 13 27(101^2) = 275427 14 82(101^2) = 836482 14 55(101^2) = 561055 14 98(101^1) = 9898 15 75(101^1) = 7575 15 36(101^1) = 3636 888888888886 2888888888884 484825981 222222222219 3196269998922 694153912 555555555551 4555555555547 855555555 888888888888 888888888888 88888888 222222222222 222222222222 22222222 555555555555 555555555555 55555555 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 28(101^15) = 325071307503 599589846747 69442028 1 57(101^15) = 661752304560 899165045164 94935557 1 86(101^15) = 998433301618 198740243582 20429086 2 71(101^14) = 8161266913 987118415530 O8709471 2 42(101^14) = 4827791695 598013710595 26278842 2 13(101^14) = 1494316477 208909005660 43848213 3 88(101^13) = 100152208 678129468772 36754488 3 76(101^13) = 86495089 312929995757 95378876 3 63(101^13) = 71699876 667297233325 67221963 4 53(101^12) = 597217 265969943951 95043653 4 6(101^12) = 67609 501807918183 23967206 4 60(101^12) = 676095 O18079181832 39672060 5 85(101^11) = 9483 180946655190 71843585 5 69(101^11) = 7698 111591990684 23025969 5 53(101^11) = 5913 O42237326177 74208353 6 5(101^10) = 5 523110627056 O2255005 6 11(101^10) = 12 150843379523 24961011 6 17(101^10) = 18 778576131990 47667017 7 100(101^9) = 1 O93685272684 36090100 7 99(101^9) = 1 O82748419957 51729199 7 99(101^9) = 1 O82748419957 51729199 8 44(101^8) = 4764569504 76355244 8 88(101^8) = 9529139009 52710488 8 31(101^8) = 3356855787 44704831 9 17(101^7) = 18226300 98581917 9 34(101^7) = 36452601 97163834 9 51(101^7) = 54678902 95745751 10 28(101^6) = 297225 64216828 10 57(101^6) = 605066 48584257 10 86(101^6) = 912907 32951686 11 98(101^5) = 10299 89849098 11 96(101^5) = 10089 69648096 11 93(101^5) = 9774 39346593 12 66(101^4) = 68 67986466 12 31(101^4) = 32 25872431 12 97(101^4) = 100 93858897 13 15(101^3) = 15454515 13 31(101^3) = 31939331 13 47(101^3) = 48424147 14 73(101^2) = 744673 14 46(101^2) = 469246 14 18(101^2) = 183618 15 62(101^1) = 6262 15 23(101^1) = 2323 15 85(101^1) 8585 333333333330 3239648060642 722368857 666666666664 2583918246703 666666666 999999999998 1856935072027 699999999 333333333333 333333333333 33333333 666666666666 666666666666 66666666 999999999999 999999999999 99999999 - B.S.Rangaswamy
S U M O F M U L T I P L E P O W E R S O F " 1 0 1 " SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 9(101^15) = 104487205983 299868165026 O4463509 1 38(101^15) = 441168203040 599443363443 29957038 1 66(101^15) = 766239510544 199033210190 99399066 2 57(101^14) = 6552003015 454447178664 99949857 2 28(101^14) = 3218527797 O65342473730 17519228 2 100(101^14) = 11494742132 376223120464 91140100 3 63(101^13) = 71699876 667297233325 67221963 3 50(101^13) = 56904664 O21664470893 39065050 3 38(101^13) = 43247544 656464997878 97689438 4 17(101^12) = 191560 255122434852 51240417 4 71(101^12) = 800045 771393698501 66945271 4 24(101^12) = 270438 O07231672732 95868824 5 95(101^11) = 10598 849293320507 27354595 5 79(101^11) = 8813 779938656000 78536979 5 63(101^11) = 7028 710583991494 29719363 6 69(101^10) = 76 218926653373 11119069 6 75(101^10) = 82 846659405840 33825075 6 81(101^10) = 89 474392158307 56531081 7 33(101^9) = 360916139985 83909733 7 32(101^9) = 349979287258 99548832 7 32(101^9) = 349979287258 99548832 8 48(101^8) = 5197712187 O1478448 8 92(101^8) = 9962281691 77833692 8 35(101^8) = 3789998469 69828035 9 39(101^7) = 41813278 73217339 9 56(101^7) = 60039579 71799256 9 74(101^7) = 79338016 O5591874 10 43(101^6) = 456453 66475843 10 72(101^6) = 764294 50843272 10 28(101^5) = 2942 82814028 11 32(101^5) = 3363 23216032 11 30(101^5) = 3153 O3015030 11 19(101^4) = 19 77147619 12 89(101^4) = 92 61375689 12 54(101^4) = 56 19261654 12 70(101^3) = 72121070 13 38(101^3) = 39151438 13 54(101^3) = 55636254 13 37(101^2) = 377437 14 91(101^2) = 928291 14 64(101^2) = 652864 14 10(101^1) = 1010 15 88(101^1) = 8888 15 49(101^1) = 4949 777777777775 2770546105037 872185903 111111111108 3111111111105 605169154 444444444441 3357977499815 641429414 777777777777 777777777777 77777777 111111111111 111111111111 11111111 444444444444 444444444444 44444444 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 19(101^15) = 220584101520 299721681721 64978519 1 47(101^15) = 545655409023 899311528469 34420547 1 76(101^15) = 882336406081 198886726886 59914076 2 14(101^14) = 1609263898 532671236865 O8759614 2 86(101^14) = 9885478233 843551883599 82380486 2 57(101^14) = 6552003015 454447178664 99949857 3 25(101^13) = 28452332 O10832235446 69532525 3 12(101^13) = 13657119 365199473014 41375612 3 42(101^12) = 473266 512655427282 67770442 4 35(101^12) = 394388 760546189402 23142035 4 89(101^12) = 1002874 276817453051 38846889 4 58(101^11) = 6470 876410658836 O1963858 5 90(101^11) = 10041 O15119987848 99599090 5 74(101^11) = 8255 945765323342 50781474 5 49(101^10) = 54 126484145149 O2099049 6 37(101^10) = 40 871018640214 56687037 6 43(101^10) = 47 498751392681 79393043 6 65(101^9) = 710895427244 83458565 7 66(101^9) = 721832279971 67819466 7 66(101^9) = 721832279971 67819466 7 84(101^8) = 9095996327 27587284 8 96(101^8) = 10395424374 O2956896 8 39(101^8) = 4223141151 94951239 8 12(101^7) = 12865624 22528412 9 78(101^7) = 83626557 46434678 9 96(101^7) = 102924993 80227296 9 43(101^6) = 456453 66475843 10 86(101^6) = 912907 32951686 10 14(101^6) = 148612 82108414 10 61(101^5) = 6411 16130561 11 65(101^5) = 6831 56532565 11 63(101^5) = 6621 36331563 11 8(101^4) = 8 32483208 12 77(101^4) = 80 12650877 12 42(101^4) = 43 70536842 12 8(101^3) = 8242408 13 77(101^3) = 79333177 13 93(101^3) = 95817993 13 27(101^2) = 275427 14 82(101^2) = 836482 14 55(101^2) = 561055 14 98(101^1) = 9898 15 75(101^1) = 7575 15 36(101^1) = 3636 888888888886 2888888888884 484825981 222222222219 3196269998922 694153912 555555555551 4555555555547 855555555 888888888888 888888888888 88888888 222222222222 222222222222 22222222 555555555555 555555555555 55555555 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 28(101^15) = 325071307503 599589846747 69442028 1 57(101^15) = 661752304560 899165045164 94935557 1 86(101^15) = 998433301618 198740243582 20429086 2 71(101^14) = 8161266913 987118415530 O8709471 2 42(101^14) = 4827791695 598013710595 26278842 2 13(101^14) = 1494316477 208909005660 43848213 3 88(101^13) = 100152208 678129468772 36754488 3 76(101^13) = 86495089 312929995757 95378876 3 63(101^13) = 71699876 667297233325 67221963 4 53(101^12) = 597217 265969943951 95043653 4 6(101^12) = 67609 501807918183 23967206 4 60(101^12) = 676095 O18079181832 39672060 5 85(101^11) = 9483 180946655190 71843585 5 69(101^11) = 7698 111591990684 23025969 5 53(101^11) = 5913 O42237326177 74208353 6 5(101^10) = 5 523110627056 O2255005 6 11(101^10) = 12 150843379523 24961011 6 17(101^10) = 18 778576131990 47667017 7 100(101^9) = 1 O93685272684 36090100 7 99(101^9) = 1 O82748419957 51729199 7 99(101^9) = 1 O82748419957 51729199 8 44(101^8) = 4764569504 76355244 8 88(101^8) = 9529139009 52710488 8 31(101^8) = 3356855787 44704831 9 17(101^7) = 18226300 98581917 9 34(101^7) = 36452601 97163834 9 51(101^7) = 54678902 95745751 10 28(101^6) = 297225 64216828 10 57(101^6) = 605066 48584257 10 86(101^6) = 912907 32951686 11 98(101^5) = 10299 89849098 11 96(101^5) = 10089 69648096 11 93(101^5) = 9774 39346593 12 66(101^4) = 68 67986466 12 31(101^4) = 32 25872431 12 97(101^4) = 100 93858897 13 15(101^3) = 15454515 13 31(101^3) = 31939331 13 47(101^3) = 48424147 14 73(101^2) = 744673 14 46(101^2) = 469246 14 18(101^2) = 183618 15 62(101^1) = 6262 15 23(101^1) = 2323 15 85(101^1) = 8585 333333333330 3239648060642 722368857 666666666664 2583918246703 666666666 999999999998 1856935072027 699999999 333333333333 333333333333 33333333 666666666666 666666666666 66666666 999999999999 999999999999 99999999 - B.S.Rangaswamy
Sum of multiple powers of palindrome prime "353" 32 digit random palindrome Sum of multiple powers of "11" - 32 digit random palindrome SN Multi power E X P A N D E D P O W E R S SN Squares E X P A N D E D P O W E R S SN Multiple power E X P A N D E D P O W E R S 1 2(353^12) = 7487321 129479627316 927293927682 A 1 616324 7639200445^2 = 37985621 462109858667 O38888198025 C 1 2(11^30) = 34898804 537772814637 117607507602 2 341(353^11) = 3616397 316080103279 139103724277 2 90681241^2 = 8223 O87469300081 2 (11^29) = 1586309 297171491574 414436704891 3 246(353^10) = 7390 641919331622 595334664054 3 7427^2 = 55160329 3 10(11^28) = 1442099 361064992340 376760640810 4 23(353^9) = 1 957492850418 334366207159 4 23^2 = 529 4 4(11^27) = 52439 976765999721 468245841484 5 274(353^8) = 66061465822 715065444144 5 3^2 = 9 5 5(11^26) = 5959 O88268863604 712300663805 6 113(353^7) = 77179397 O13779723281 37985621 462109866890 126412658973 6 10(11^23) = 8 954302432552 373722465310 7 285(353^6) = 551433 431495580765 37985621 462109866890 126412658973 7 3(11^22) = 244208248160 519283339963 8 332(353^5) = 1819 749508041676 8 3(11^20) = 2018249984 797680027603 9 77(353^4) = 1 195610021837 SN Squares & Cubes E X P A N D E D P O W E R S 9 8(11^19) = 489272723 587316370328 10 217(353^3) = 9545174009 B 1 3 3615513156^3 = 37985621 459853960535 381468900416 10 8(11^18) = 44479338 507937851848 11 69(353^2) = 8598021 2 4 7496382543^2 = 2255906354 670995146849 11 5(11^17) = 2527235 142496468855 12 12(353^1) = 4236 3 271934^2 = 73948100356 12 10(11^16) = 459497 298635721610 11111109 2111111111107 4097331387860 4 715^2 = 511225 13 8(11^15) = 33417 985355325208 11111111 111111111111 111111111111 5 10^2 = 100 14 5(11^14) = 1898 749167916205 6 3^3 = 27 15 5(11^13) = 172 613560719655 37985621 462109866889 1126412658973 16 9(11^12) = 28 245855390489 On similar lines, the other EIGHT repdigital palindromes of 37985621 462109866890 126412658973 17 5(11^11) = 1 426558353055 18 8(11^10) = 207499396808 32 digits can be generated from multiple powers of " 353 " 19 4(11^9) = 9431790764 The 32 digit random palindrome is expressed as sum of 20 9(11^8) = 1929229929 a palindrome prime. 21 4(11^6) = 7086244 A. Squares 22 3(11^5) = 483153 23 (11^4) = 14641 B. Squares and Cubes 24 5(11^3) = 6655 25 6(11^2) = 726 C. Multiple powers of "11" the lowest palindrome prime. 26 10(11^1) = 110 37985618 3373841003277 8555789317751 37985621 462109866890 126412658973
3rd September 2009 Dear Mr Patrick Hope the temperature is coming down appreciably and you are getting into good health and spirits. Please find herewith enclosed all repdigital palindromes of 32 digits, generated by multiple powers of palindrome prime “353” at Excel sheet 2 of enclosure for your kind perusal. Next to PP 101, this is the only palindrome prime in 3, 4 and 5 digits, which is capable of generating repdigitals of 32 digits. By coincidence ‘353’ also happens to be my house number. Also enclosed at sheet 3, the random palindrome of 32 digits, generated by A. Squares B. Squares and Cubes C. Multiple powers of second lowest palindrome prime “11” D. Multiple powers of prime “5503” Next to 11, 5503 is the only number in 2, 3 and 4 digits, which is capable of generating this particular Random palindrome of 32 digits! This concludes my submissions to you on the subject – Repdigitals & Random palindrome as Sum of multiple powers of primes. Thanking You With Regards B.S.Rangaswamy
S U M O F M U L T I P L E P O W E R S O F " 1 0 1 " SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 9(101^15) = 104487205983 299868165026 O4463509 1 38(101^15) = 441168203040 599443363443 29957038 1 66(101^15) = 766239510544 199033210190 99399066 2 57(101^14) = 6552003015 454447178664 99949857 2 28(101^14) = 3218527797 O65342473730 17519228 2 100(101^14) = 11494742132 376223120464 91140100 3 63(101^13) = 71699876 667297233325 67221963 3 50(101^13) = 56904664 O21664470893 39065050 3 38(101^13) = 43247544 656464997878 97689438 4 17(101^12) = 191560 255122434852 51240417 4 71(101^12) = 800045 771393698501 66945271 4 24(101^12) = 270438 O07231672732 95868824 5 95(101^11) = 10598 849293320507 27354595 5 79(101^11) = 8813 779938656000 78536979 5 63(101^11) = 7028 710583991494 29719363 6 69(101^10) = 76 218926653373 11119069 6 75(101^10) = 82 846659405840 33825075 6 81(101^10) = 89 474392158307 56531081 7 33(101^9) = 360916139985 83909733 7 32(101^9) = 349979287258 99548832 7 32(101^9) = 349979287258 99548832 8 48(101^8) = 5197712187 O1478448 8 92(101^8) = 9962281691 77833692 8 35(101^8) = 3789998469 69828035 9 39(101^7) = 41813278 73217339 9 56(101^7) = 60039579 71799256 9 74(101^7) = 79338016 O5591874 10 43(101^6) = 456453 66475843 10 72(101^6) = 764294 50843272 10 28(101^5) = 2942 82814028 11 32(101^5) = 3363 23216032 11 30(101^5) = 3153 O3015030 11 19(101^4) = 19 77147619 12 89(101^4) = 92 61375689 12 54(101^4) = 56 19261654 12 70(101^3) = 72121070 13 38(101^3) = 39151438 13 54(101^3) = 55636254 13 37(101^2) = 377437 14 91(101^2) = 928291 14 64(101^2) = 652864 14 10(101^1) = 1010 15 88(101^1) = 8888 15 49(101^1) = 4949 777777777775 2770546105037 872185903 111111111108 3111111111105 605169154 444444444441 3357437499815 641429414 777777777777 777777777777 77777777 111111111111 111111111111 11111111 444444444444 444444444444 44444444 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 19(101^15) = 220584101520 299721681721 64978519 1 47(101^15) = 545655409023 899311528469 34420547 1 76(101^15) = 882336406081 198886726886 59914076 2 14(101^14) = 1609263898 532671236865 O8759614 2 86(101^14) = 9885478233 843551883599 82380486 2 57(101^14) = 6552003015 454447178664 99949857 3 25(101^13) = 28452332 O10832235446 69532525 3 12(101^13) = 13657119 365199473014 41375612 3 42(101^12) = 473266 512655427282 67770442 4 35(101^12) = 394388 760546189402 23142035 4 89(101^12) = 1002874 276817453051 38846889 4 58(101^11) = 6470 876410658836 O1963858 5 90(101^11) = 10041 O15119987848 99599090 5 74(101^11) = 8255 945765323342 50781474 5 49(101^10) = 54 126484145149 O2099049 6 37(101^10) = 40 871018640214 56687037 6 43(101^10) = 47 498751392681 79393043 6 65(101^9) = 710895427244 83458565 7 66(101^9) = 721832279971 67819466 7 66(101^9) = 721832279971 67819466 7 84(101^8) = 9095996327 27587284 8 96(101^8) = 10395424374 O2956896 8 39(101^8) = 4223141151 94951239 8 12(101^7) = 12865624 22528412 9 78(101^7) = 83626557 46434678 9 96(101^7) = 102924993 80227296 9 43(101^6) = 456453 66475843 10 86(101^6) = 912907 32951686 10 14(101^6) = 148612 82108414 10 61(101^5) = 6411 16130561 11 65(101^5) = 6831 56532565 11 63(101^5) = 6621 36331563 11 8(101^4) = 8 32483208 12 77(101^4) = 80 12650877 12 42(101^4) = 43 70536842 12 8(101^3) = 8242408 13 77(101^3) = 79333177 13 93(101^3) = 95817993 13 27(101^2) = 275427 14 82(101^2) = 836482 14 55(101^2) = 561055 14 98(101^1) = 9898 15 75(101^1) = 7575 15 36(101^1) = 3636 888888888886 2888888888884 484825981 222222222219 3196269998922 694153912 555555555551 4555555555547 855555555 888888888888 888888888888 88888888 222222222222 222222222222 22222222 555555555555 555555555555 55555555 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 28(101^15) = 325071307503 599589846747 69442028 1 57(101^15) = 661752304560 899165045164 94935557 1 86(101^15) = 998433301618 198740243582 20429086 2 71(101^14) = 8161266913 987118415530 O8709471 2 42(101^14) = 4827791695 598013710595 26278842 2 13(101^14) = 1494316477 208909005660 43848213 3 88(101^13) = 100152208 678129468772 36754488 3 76(101^13) = 86495089 312929995757 95378876 3 63(101^13) = 71699876 667297233325 67221963 4 53(101^12) = 597217 265969943951 95043653 4 6(101^12) = 67609 501807918183 23967206 4 60(101^12) = 676095 O18079181832 39672060 5 85(101^11) = 9483 180946655190 71843585 5 69(101^11) = 7698 111591990684 23025969 5 53(101^11) = 5913 O42237326177 74208353 6 5(101^10) = 5 523110627056 O2255005 6 11(101^10) = 12 150843379523 24961011 6 17(101^10) = 18 778576131990 47667017 7 100(101^9) = 1 O93685272684 36090100 7 99(101^9) = 1 O82748419957 51729199 7 99(101^9) = 1 O82748419957 51729199 8 44(101^8) = 4764569504 76355244 8 88(101^8) = 9529139009 52710488 8 31(101^8) = 3356855787 44704831 9 17(101^7) = 18226300 98581917 9 34(101^7) = 36452601 97163834 9 51(101^7) = 54678902 95745751 10 28(101^6) = 297225 64216828 10 57(101^6) = 605066 48584257 10 86(101^6) = 912907 32951686 11 98(101^5) = 10299 89849098 11 96(101^5) = 10089 69648096 11 93(101^5) = 9774 39346593 12 66(101^4) = 68 67986466 12 31(101^4) = 32 25872431 12 97(101^4) = 100 93858897 13 15(101^3) = 15454515 13 31(101^3) = 31939331 13 47(101^3) = 48424147 14 73(101^2) = 744673 14 46(101^2) = 469246 14 18(101^2) = 183618 15 62(101^1) = 6262 15 23(101^1) = 2323 15 85(101^1) 8585 333333333330 3239648060642 722368857 666666666664 2583918246703 666666666 999999999998 1856935072027 699999999 333333333333 333333333333 33333333 666666666666 666666666666 66666666 999999999999 999999999999 99999999 - B.S.Rangaswamy
Repdigital palindromes as sum of multiple powers of pal prime "353" SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 2(353^12) = 7487321 129479627316 927293927682 1 11(353^12) = 41180266 212137950243 100116602251 1 20(353^12) = 74873211 294796273169 272939276820 2 341(353^11) = 3616397 316080103279 139103724277 2 307(353^11) = 3255818 111544257204 386231212179 2 273(353^11) = 2895238 907008411129 633358700081 3 246(353^10) = 7390 641919331622 595334664054 3 278(353^10) = 8352 O26234041427 160581449622 3 310(353^10) = 9313 410548751231 725828235190 4 23(353^9) = 1 957492850418 334366207159 4 95(353^9) = 8 O85296556075 728903899135 4 166(353^9) = 14 127991876932 326295234278 5 274(353^8) = 66061465822 715065444144 5 38(353^8) = 9161809128 697709806118 5 155(353^8) = 37370537235 477500524955 6 113(353^7) = 77179397 O13779723281 6 102(353^7) = 69666358 366420635174 6 90(353^7) = 61470316 205665266330 7 285(353^6) = 551433 431495580765 7 84(353^6) = 162527 748230276436 7 236(353^6) = 456625 578361252844 8 332(353^5) = 1819 749508041676 8 269(353^5) = 1474 435595371117 8 207(353^5) = 1134 602855917551 9 77(353^4) = 1 195610021837 9 310(353^4) = 4 813494893110 9 190(353^4) = 2 950206547390 10 217(353^3) = 9545174009 10 162(353^3) = 7125890274 10 108(353^3) = 4750593516 11 69(353^2) = 8598021 11 276(353^2) = 34392084 11 130(353^2) = 16199170 12 12(353^1) = 4236 12 48(353^1) = 16944 12 84(353^1) = 29652 11111109 2111111111107 4097331387860 44444444 332913846938 4444444444444 77777776 1777777777773 4777777777777 11111111 111111111111 111111111111 44444444 444444444444 444444444444 77777777 777777777777 777777777777 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 5(353^12) = 18718302 823699068292 318234819205 1 14(353^12) = 52411247 906357391218 491057493774 1 23(353^12) = 86104192 989015714144 663880168343 2 330(353^11) = 3499739 338142035431 424939088010 2 296(353^11) = 3139160 133606189356 672066575912 2 262(353^11) = 2778580 929070343281 919194063814 3 139(353^10) = 4176 O13117020713 580290724811 3 171(353^10) = 5137 397431730518 145537510379 3 203(353^10) = 6098 781746440322 710784295947 4 47(353^9) = 4 O00094085637 465878771151 4 118(353^9) = 10 O42789406494 O63270106294 4 190(353^9) = 16 170593112151 457807798270 5 195(353^8) = 47014546844 632984531395 5 312(353^8) = 75223274951 412775250232 5 76(353^8) = 18323618257 395419612236 6 227(353^7) = 155041797 540955727299 6 216(353^7) = 147528758 893596639192 6 204(353^7) = 139332716 732841270348 7 218(353^6) = 421798 203740479322 7 17(353^6) = 32892 520475174993 7 169(353^6) = 326990 350606151401 8 311(353^5) = 1704 644870484823 8 249(353^5) = 1364 812131031257 8 186(353^5) = 1019 498218360698 9 155(353^4) = 2 406747446555 9 35(353^4) = 543459100835 9 267(353^4) = 4 145816569227 10 81(353^3) = 3562945137 10 26(353^3) = 1143661402 10 325(353^3) = 14295767525 11 138(353^2) = 17196042 11 345(353^2) = 42990105 11 199(353^2) = 24797191 12 24(353^1) = 8472 12 60(353^1) = 21180 12 96(353^1) = 33888 22222221 1209011115868 4222222222222 55555554 1512766149057 4492285449261 88888886 2888888888884 4888888888888 22222222 222222222222 222222222222 55555555 555555555555 555555555555 88888888 888888888888 888888888888 SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S SN Multi power E X P A N D E D P O W E R S 1 8(353^12) = 29949284 517918509267 709175710728 1 17(353^12) = 63642229 600576832193 881998385297 1 26(353^12) = 97335174 683235155120 O54821059866 2 319(353^11) = 3383081 360203967583 710774451743 2 285(353^11) = 3022502 155668121508 957901939645 2 251(353^11) = 2661922 951132275434 205029427547 3 32(353^10) = 961 384314709804 565246785568 3 64(353^10) = 1922 768629419609 130493571136 3 96(353^10) = 2884 152944129413 695740356704 4 71(353^9) = 6 O42695320856 597391335143 4 142(353^9) = 12 O85390641713 194782670286 4 213(353^9) = 18 128085962569 792174005429 5 116(353^8) = 27967627866 550903618676 5 233(353^8) = 56176355973 330694337513 5 350(353^8) = 84385084080 110485056350 6 341(353^7) = 232904198 O68131731317 6 329(353^7) = 224708155 907376362473 6 318(353^7) = 217195117 260017274366 7 151(353^6) = 292162 975985377879 7 303(353^6) = 586260 806116354287 7 102(353^6) = 197355 122851049958 8 290(353^5) = 1589 540232927970 8 228(353^5) = 1249 707493474404 8 165(353^5) = 904 393580803845 9 232(353^4) = 3 602357468392 9 112(353^4) = 1 739069122672 9 345(353^4) = 5 356953993945 10 298(353^3) = 13108119146 10 244(353^3) = 10732822388 10 189(353^3) = 8313538653 11 207(353^2) = 25794063 11 61(353^2) = 7601149 11 268(353^2) = 33395212 12 36(353^1) = 12708 12 72(353^1) = 25416 12 108(353^1) = 38124 33333332 1290638012472 5265201602016 66666665 1581276024948 5666666666666 99999998 1999999999997 2945178940133 33333333 333333333333 333333333333 66666666 666666666666 666666666666 99999999 999999999999 999999999999 - B.S.Rangaswamy
32 digit random palindrome Sum of multiple powers of "11" - 32 digit random palindrome Sum of multiple powers of prime "5503" - Random palindrome SN Squares E X P A N D E D P O W E R S SN Multiple power E X P A N D E D P O W E R S SN Multiple power E X P A N D E D P O W E R S A 1 616324 7639200445^2 = 37985621 462109858667 O38888198025 C 1 2(11^30) = 34898804 537772814637 117607507602 1 45(5503^8) = 37845009 297226706356 O60165555245 2 90681241^2 = 8223 O87469300081 2 (11^29) = 1586309 297171491574 414436704891 2 920(5503^7) = 140599 707446235668 817313192040 3 7427^2 = 55160329 3 10(11^28) = 1442099 361064992340 376760640810 3 448(5503^6) = 12 441567235245 909272838592 4 23^2 = 529 4 4(11^27) = 52439 976765999721 468245841484 4 3144(5503^5) = 15866462200 655624023992 5 3^2 = 9 5 5(11^26) = 5959 O88268863604 712300663805 5 3519(5503^4) = 3227136 371377071039 37985621 462109866890 126412658973 6 10(11^23) = 8 954302432552 373722465310 6 1694(5503^3) = 282 300693104738 37985621 462109866890 126412658973 7 3(11^22) = 244208248160 519283339963 7 395(5503^2) = 11961788555 8 3(11^20) = 2018249984 797680027603 8 924(5503^1) = 5084772 SN Squares & Cubes E X P A N D E D P O W E R S 9 8(11^19) = 489272723 587316370328 37985620 1462109866887 3066247103728 B 1 3 3615513156^3 = 37985621 459853960535 381468900416 10 8(11^18) = 44479338 507937851848 37985621 462109866890 126412658973 2 4 7496382543^2 = 2255906354 670995146849 11 5(11^17) = 2527235 142496468855 3 271934^2 = 73948100356 12 10(11^16) = 459497 298635721610 4 715^2 = 511225 13 8(11^15) = 33417 985355325208 5 10^2 = 100 14 5(11^14) = 1898 749167916205 6 3^3 = 27 15 5(11^13) = 172 613560719655 37985621 462109866889 1126412658973 16 9(11^12) = 28 245855390489 37985621 462109866890 126412658973 17 5(11^11) = 1 426558353055 18 8(11^10) = 207499396808 19 4(11^9) = 9431790764 The 32 digit random palindrome is expressed as sum of 20 9(11^8) = 1929229929 21 4(11^6) = 7086244 A. Squares 22 3(11^5) = 483153 23 (11^4) = 14641 B. Squares and Cubes 24 5(11^3) = 6655 25 6(11^2) = 726 C. Multiple powers of "11" the lowest palindrome prime. 26 10(11^1) = 110 37985618 3373841003277 8555789317751 D. Multiple powers of prime " 5503 ". 37985621 462109866890 126412658973 - B.S.Rangaswamy
Dt 18th September 2009 Dear Mr Patrick SUB: Palindrome as sum of Powers – WON 178 63 digit palindrome a dream sum of different powers as prescribed in your WON Plate 178, has now become a reality! This Palindrome is: 34336838202925124846578490892810182980948756484215292028386334 which is the sum of 21 different powers, the degree of powers ranging from 2 to 70 and is depicted in the following list: -------------------------------------------------------------- 1. 90^32 12. 7^17 2. 48^18 13. 93^7 3. 14^23 14. 67^6 4. 19^20 15. 5^13 5. 3^50 16. 2^27 6. 33^15 17. 256^3 7. 2^70 18. 13^4 8. 121^10 19. 2^12 9. 2^67 20. 43^2 10. 272^8 21. 2^5 11. 79^9 --------------------------------------------------------------- There also exists a higher value 63 digit palindrome, which happens to be the sum of 22 different powers!! These are not sporadic, but engineered by an Engineer. I shall be submitting details of both these sums in Excel format, when complete for your perusal. Kindly bear with this requisite and reasonable delay. Thanking You With High Regards B.S.Rangaswamy
Dear Mr Patrick Extremely sorry for missing '3', which is now included at the end of 63 digit palindrome. - B.S.Rangaswamy
Dt 18th September 2009 Dear Mr Patrick SUB: Palindrome as sum of Powers – WON 178 63 digit palindrome a dream sum of different powers as prescribed in your WON Plate 178, has now become a reality! This Palindrome is: 343368382029251248465784908928101829809487564842152920283863343 which is the sum of 21 different powers, the degree of powers ranging from 2 to 70 and is depicted in the following list: -------------------------------------------------------------- 1. 90^32 12. 7^17 2. 48^18 13. 93^7 3. 14^23 14. 67^6 4. 19^20 15. 5^13 5. 3^50 16. 2^27 6. 33^15 17. 256^3 7. 2^70 18. 13^4 8. 121^10 19. 2^12 9. 2^67 20. 43^2 10. 272^8 21. 2^5 11. 79^9 --------------------------------------------------------------- There also exists a higher value 63 digit palindrome, which happens to be the sum of 22 different powers!! These are not sporadic, but engineered by an Engineer. I shall be submitting details of both these sums in Excel format, when complete for your perusal. Kindly bear with this requisite and reasonable delay. Thanking You With High Regards B.S.Rangaswamy
Dt 22nd September 2009 Dear Mr Patrick SUB: Palindromes as sum of Powers - WON 178 Further to my submissions dated 18th September, the second 63 digit palindrome is: 343368382029251248465784908928111829809487564842152920283863343 ( Sheet 2 of Excel format; Earlier palindrome at Sheet 1 ) which is the sum of 22 different powers, and is illustrated in the following list with only the addition of 10^31 to the first: ------------------------------------------------------------- 1. 90^32 12. 79^9 2. 10^31 13. 7^17 3. 48^18 14. 93^7 4. 14^23 15. 67^6 5. 19^20 16. 5^13 6. 3^50 17. 2^27 7. 33^15 18. 256^3 8. 2^70 19. 13^4 9. 121^10 20. 2^12 10. 2^67 21. 43^2 11. 272^8 22. 2^5 --------------------------------------------------------------- Various constituent powers for the following palindromes may be worked out by enthusiastic readers: 343368382029251248465784908928121829809487564842152920283863343 343368382029251248465784908928131829809487564842152920283863343 3433 - - - - - - - - - - - - - - - - - - - - - - 141 - - - - - - - - - - - - - - - - - - - - - -3343 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 343368382029251248465784908928191829809487564842152920283863343 Arriving at the constituent powers for the following 63 digit palindrome can turn out to be highly daunting / haunting task to the involved reader and likewise for the next lower values of 63 digit palindromes: 343368382029251248465784908928090829809487564842152920283863343 Grandson’s easy and harmonic way of calculating high degree powers was of great help. He is now nearer to you in England for higher studies. It was a daunting experience to search and get at the last few appropriate powers to arrive at the first 63 digit palindrome. Thank You immensely for bestowing me an opportunity to participate in this highly interesting and educating topic. With High Regards B.S.Rangaswamy
63 Digit Palindrome - Sum of Powers 63 Digit Palindrome - Sum of Powers A POWERS A SN Powers E X P A N D E D P O W E R S ORDER SORT SN Powers E X P A N D E D P O W E R S 1 90^32 343368382029251 248465784908 928100000000 O00000000000 O00000000000 32 70 1 90^32 343368382029251 248465784908 928100000000 O00000000000 O00000000000 2 48^18 1829541 532030568071 946613817344 18 67 2 48^18 1829541 532030568071 946613817344 3 14^23 229 585692886981 495482220544 23 50 3 14^23 229 585692886981 495482220544 4 19^20 37 589973457545 958193355601 20 32 4 19^20 37 589973457545 958193355601 5 3^50 717897987691 852588770249 50 27 5 3^50 717897987691 852588770249 6 33^15 59938945498 865420543457 15 23 6 33^15 59938945498 865420543457 7 2^70 1180591620 717411303424 70 20 7 2^70 1180591620 717411303424 8 121^10 672749994 932560009201 10 18 8 121^10 672749994 932560009201 9 2^67 147573952 589676412928 67 17 9 2^67 147573952 589676412928 10 272^8 29960650 O73923649536 8 15 10 272^8 29960650 O73923649536 11 79^9 119851 595982618319 9 13 11 79^9 119851 595982618319 12 7^17 232 630513987207 17 12 12 7^17 232 630513987207 13 93^7 60 170087060757 7 10 13 93^7 60 170087060757 14 67^6 90458382169 6 9 14 67^6 90458382169 15 5^13 1220703125 13 8 15 5^13 1220703125 16 2^27 134217728 27 7 16 2^27 134217728 17 256^3 16777216 3 6 17 256^3 16777216 18 13^4 28561 4 5 18 13^4 28561 19 2^12 4096 12 4 19 2^12 4096 20 43^2 1849 2 3 20 43^2 1849 21 2^5 32 5 2 21 2^5 32 343368382029251 248465784908 928101829807 2487564842145 7920283863343 343368382029251 248465784908 928101829807 2487564842145 7920283863343 < Total of individual columns 343368382029251 248465784908 928101829809 487564842152 920283863343 < 63 Digit Palindrome 343368382029251 248465784908 928101829809 487564842152 920283863343 < 63 Digit Palindrome - B.S.Rangaswamy 21.09.2009
63 Digit Palindrome - Sum of Powers 63 Digit Palindrome - Sum of Powers B POWERS B SN Powers E X P A N D E D P O W E R S ORDER SORT SN Powers E X P A N D E D P O W E R S 1 90^32 343368382029251 248465784908 928100000000 O00000000000 O00000000000 32 70 1 90^32 343368382029251 248465784908 928100000000 O00000000000 O00000000000 Add 2 10^31 10000000 O00000000000 O00000000000 31 67 Add 2 10^31 10000000 O00000000000 O00000000000 to A 3 48^18 1829541 532030568071 946613817344 18 50 to A 3 48^18 1829541 532030568071 946613817344 4 14^23 229 585692886981 495482220544 23 32 4 14^23 229 585692886981 495482220544 5 19^20 37 589973457545 958193355601 20 31 5 19^20 37 589973457545 958193355601 6 3^50 717897987691 852588770249 50 27 6 3^50 717897987691 852588770249 7 33^15 59938945498 865420543457 15 23 7 33^15 59938945498 865420543457 8 2^70 1180591620 717411303424 70 20 8 2^70 1180591620 717411303424 9 121^10 672749994 932560009201 10 18 9 121^10 672749994 932560009201 10 2^67 147573952 589676412928 67 17 10 2^67 147573952 589676412928 11 272^8 29960650 O73923649536 8 15 11 272^8 29960650 O73923649536 12 79^9 119851 595982618319 9 13 12 79^9 119851 595982618319 13 7^17 232 630513987207 17 12 13 7^17 232 630513987207 14 93^7 60 170087060757 7 10 14 93^7 60 170087060757 15 67^6 90458382169 6 9 15 67^6 90458382169 16 5^13 1220703125 13 8 16 5^13 1220703125 17 2^27 134217728 27 7 17 2^27 134217728 18 256^3 16777216 3 6 18 256^3 16777216 19 13^4 28561 4 5 19 13^4 28561 20 2^12 4096 12 4 20 2^12 4096 21 43^2 1849 2 3 21 43^2 1849 22 2^5 32 5 2 22 2^5 32 343368382029251 248465784908 928111829807 2487564842145 7920283863343 343368382029251 248465784908 928111829807 2487564842145 7920283863343 < Total of individual columns 343368382029251 248465784908 928111829809 487564842152 920283863343 < 63 Digit Palindrome 343368382029251 248465784908 928111829809 487564842152 920283863343 < 63 Digit Palindrome - B.S.Rangaswamy 21.09.2009


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B.S. Rangaswamy (email) dd. [ various dates in 2009 ].






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