World!Of
Numbers  WON plate
178 |

[ May 8, 2009 ] [ Last Update March 4, 2010 ]
Searching for ever larger sporadic palindromes
of the form Ap + Bq (Sum Of Powers)
Patrick De Geest

Let us go fishing palindromes in a vast unlimited pool of numbers.
The rod we use can be for example this formula allowing a freedom
of four variables - two basenumbers A & B and two exponents p & q,
which need not to be consecutive necessarily.

Ap + Bq = palindrome
or with more terms...
Ap + Bq + Cr + ... = palindrome [ all variables > 1 ]

So thousands or even millions palindromes could be generated.
The choice of these variables is all up to you. Define your own
strategy. Apply the most obscure number theory if you like.
Of course we are mainly interested in the sporadic record solutions
[ sporadic means exclusion of trivial solutions or infinite patterns that might arise ]
that once in a while will pop up on your screen. I am more than happy
to collect and display these record palindromes in the tables below
giving credit to the rightful claimant i.e., its discoverer.
I'll try to maintain a top 100 - or so - of these submissions.

If you spot some beautiful or interesting phenomena/curios but
are not directly record palindromes, report them anyway and I
will publish these as well in a separate section. Palindromic
primes
(or palprimes) will be colormarked in their cells resp. Here are some appetizers !

The powers expressionThe record palindromesLengthDiscoverer
3642 + 4390462838463127920454029721364823pdg
6422 + 295509232580554398893455085220pdg
103 + 1010141041016191610140117pdg
21004 + 33252348167761843214pdg Here are your contributions !

Palindrome (63 digits) The 9 powers [ February 28, 2010 ] B.S.Rangaswamy encloses details of his new finding of a 63 digit palindrome constituted out of nine different powers, which fully meet the requirement of this WON plate. 343368382029251248465784908928151829809487564842152920283863343 1. 63 digits 9032 + 2. 32 digits 22025005 + 3. 25 digits 729 + 4. 23 digits 14087 + 5. 19 digits 547714 + 6. 13 digits 1710 + 7. 11 digits 159 + 8. 10 digits 16593 + 9. 10 digits 590702 " This is definitely not sporadic, but strategically engineered to the core by an Engineer. 'Start with palindrome ' is the slogan adopted. I wish others to attempt and find palindromes having higher number of digits and lesser number of constituent powers. I was thrilled and educated numerically, while working on this most scintillating exercise and highly thankful for bestowing me this opportunity."

B.S.Rangaswamy submitted earlier two 63 digit
palindromes comprising of 14 & 13 different constituent powers.
I will present these (and much more material) later in a separate
page dedicated to this devoted contributor.
See the webpage Palindromic Sums of Powers

Palindrome B.S.Rangaswamy's grandson [ July 29, 2009 ] dictated this 32 digits long random palindrome expressed as a summation of powers in two ways ! 37985621462109866890126412658973    (32 digits) 61632476392004452 + 906812412 + 74272 + 232 + 32 OR 336155131563 + 474963825432 + 2719342 + 7152 + 102 + 33 With the liberty to use squares, cubes and other powers more than once,every palindrome can be expressed as a summation of powers in several ways !Lowest palindrome 11 is lone exception to this statement. All numbers morethan 23 are either powers or sums of powers. To arrive at the least numberof constituent powers is really an intellectual task.

The powers expressionThe palindromic curioLen
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

The powers expressionThe palindromes/palprimesLenExponents
Jeff Heleen [ June 20, 2009 ] gives a few largest solutions with four summands.
He used A^p + B^q + C^r + D^s and
A,B,C and D were restricted to values from 2 to 1000.
10005 + 9754 + 4403 + 1242100090377309000116{5,4,3,2}
9995 + 9664 + 4603 + 792299588086808859915
9986 + 8335 + 3944 + 346398846093883906488918{6,5,4,3}
6786 + 7475 + 9024 + 82839736879989978637917
9827 + 7066 + 185 + 894488072734602064372708821{7,6,5,4}
7537 + 5936 + 1745 + 419413731007039307001373121
Jeff Heleen [ June 29, 2009 ] gives a few more solutions with two summands.
They don't necessarily have any limits except for the lower powers,
otherwise it would take years to run the program.
95014232 + 14506929420466766402492919{2,6}
99944872 + 248633254353534523315
18272282 + 2152721374495305775035944731224{2,7}
34091492 + 118733016968696103315
47490222 + 2198529120721612702192519{2,8}
16432272 + 408925379497352913
20000012 + 1009100000400000400000119{2,9}
28071982 + 39788036063088713
6867142 + 49107979273787372979717{2,10}
65307652 + 241010605427245060115
4426143 + 9458671125595521176817{3,5}
6613 + 5859451615499
11684 + 2526625977663113113667795221{4,6}
10014 + 106100400700400113
380484 + 1748293591328182319539219{4,8}
100014 + 1081000400070004000117

The powers expressionThe palindromesLengthDiscoverer
Jean Claude Rosa [ June 20, 2009 ] searched for solutions including
the number of the beast 666.
6662 + 13663254933945210JCR
6663 + 88739932723999JCR
6663 + 16091980334225895183107013815985221JCR

```

``` Aficionado's of Ubasic might like to run my little program code.
[ ps. alas, Ubasic doesn't work with a 64-bit engine like Vista. ]
It is free for use. Amend or improve or optimize it as you like. If
you wrote other/faster code and/or in other languages that you like
to share with my readers please send it in and I will make it public.

```
10   color 15:cls
20   'file SUMPABPQ.UB by Patrick De Geest
30   Ti="Palindromes of the form A^p+B^q"
40   print Ti:print
50   open "sumpower.txt" for append as #1
60   print #1,"********** ********** ********** **********"
70   A=2:B=0:Cc=1
80   input "exponent p ";EP
90   input "exponent q ";EQ
100   input "max length palindrome ";MP
110   loop
120   X=A^EP+B^EQ
130   Q=str(X):L=len(Q)
140   for P=2 to L
150   if mid(Q,P)<>mid(Q,L+2-P) then cancel for:goto 200
160   next P
170   M=str(A)+" ^"+str(EP)+" +"+str(B)+" ^"+str(EQ)+" = "+
str(X)+" ["+str(alen(X))+" ]"
180   print #1,M
190   beep:print Cc;spc(6);A;"^";EP;" +";B;"^";EQ;" = ";:
color 10:print X;:color 15:print " [";alen(X);"]":inc Cc
200   inc B
210   if alen(X)>MP then color 11:print A;"|";B;chr(13);:
color 15:inc A:B=0
220   endloop
230   ' use close #1 after Ctrl+C or Ctrl+Break
``` Contributors
B.S.Rangaswamy (email)
Jeff Heleen (email)
Jean Claude Rosa (email) ```

``` A000178 Prime Curios! Prime Puzzle  Wikipedia 178 Le nombre 178 ```

```

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Patrick De Geest - Belgium - Short Bio - Some Pictures