Index Nr 
Base Sequence Expression  
Various Palindromic Sums  Length 
  
Multigrade Palprimes Patterns Contribution by Carlos Rivera [ August 13, 1999 ] See also Carlos' Puzzle 65 on Multigrade Relations. 
2 
10501 + 14741 + 15451 = 11411 + 12721 + 16561
10501^{2} + 14741^{2} + 15451^{2} = 11411^{2} + 12721^{2} + 16561^{2}


1 
181 + 727 + 757 = 353 + 383 + 929
181^{2} + 727^{2} + 757^{2} = 353^{2} + 383^{2} + 929^{2}


! 
Next is an ingenuously beautiful trigrade equation using palindromes
from Albert Beiler's book "Recreations in the Theory of Numbers" 13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 330330 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097
13031^{2} + 42024^{2} + 53035^{2} + 57075^{2} + 68086^{2} + 97079^{2} = 22066126024 = 31013^{2} + 24042^{2} + 35053^{2} + 75057^{2} + 86068^{2} + 79097^{2}
13031^{3} + 42024^{3} + 53035^{3} + 57075^{3} + 68086^{3} + 97079^{3} = 1642056213257460 = 31013^{3} + 24042^{3} + 35053^{3} + 75057^{3} + 86068^{3} + 79097^{3}

Trigrade from A. Beiler's book page 164 
  
Sums of Squares of Consecutive Odd Numbers 11 terms Entry 1 by Kimberly Pellechi [ July 1, 2003 ] Entry 2 by Hugo Sánchez [ March 17, 1999 ] Entries 3 to 17 by Kimberly Pellechi [ July 25, 2003 ] 
17 
4228461^{2} + 4228463^{2} + ... + 4228479^{2} + 4228481^{2}  KP 
196.679.636.976.691  15 
16 
1011109^{2} + 1011111^{2} + ... + 1011127^{2} + 1011129^{2}  KP 
11.245.977.954.211  14 
15 
424669^{2} + 424671^{2} + ... + 424687^{2} + 424689^{2}  KP 
1.983.874.783.891  13 
14 
124951^{2} + 124953^{2} + ... + 124969^{2} + 124971^{2}  KP 
171.767.767.171  12 
13 
119091^{2} + 119093^{2} + ... + 119109^{2} + 119111^{2}  KP 
156.035.530.651  12 
12 
101109^{2} + 101111^{2} + ... + 101127^{2} + 101129^{2}  KP 
112.475.574.211  12 
11 
42419^{2} + 42421^{2} + ... + 42437^{2} + 42439^{2}  KP 
19.802.420.891  11 
10 
13161^{2} + 13163^{2} + ... + 13179^{2} + 13181^{2}  KP 
1.908.228.091  10 
9 
11081^{2} + 11083^{2} + ... + 11099^{2} + 11101^{2}  KP 
1.353.113.531  10 
8 
10121^{2} + 10123^{2} + ... + 10139^{2} + 10141^{2}  KP 
1.129.009.211  10 
7 
10109^{2} + 10111^{2} + ... + 10127^{2} + 10129^{2}  KP 
1.126.336.211  10 
6 
1191^{2} + 1193^{2} + 1195^{2} + 1197^{2} + 1199^{2} + 1201^{2} + 1203^{2} + 1205^{2} + 1207^{2} + 1209^{2} + 1211^{2}  KP 
15.866.851  8 
5 
1009^{2} + 1011^{2} + 1013^{2} + 1015^{2} + 1017^{2} + 1019^{2} + 1021^{2} + 1023^{2} + 1025^{2} + 1027^{2} + 1029^{2}  KP 
11.422.411  8 
4 
99^{2} + 101^{2} + 103^{2} + 105^{2} + 107^{2} + 109^{2} + 111^{2} + 113^{2} + 115^{2} + 117^{2} + 119^{2}  KP 
131.131  6 
3 
29^{2} + 31^{2} + 33^{2} + 35^{2} + 37^{2} + 39^{2} + 41^{2} + 43^{2} + 45^{2} + 47^{2} + 49^{2}  KP 
17.171  5 
2 
21^{2} + 23^{2} + 25^{2} + 27^{2} + 29^{2} + 31^{2} + 33^{2} + 35^{2} + 37^{2} + 39^{2} + 41^{2}  HS 
11.011  5 
1 
1^{2} + 3^{2} + 5^{2} + 7^{2} + 9^{2} + 11^{2} + 13^{2} + 15^{2} + 17^{2} + 19^{2} + 21^{2}  KP 
1.771  4 
  
Sums of Squares of Consecutive Odd Numbers 9 terms Entry 1 by Hugo Sánchez [ March 17, 1999 ] Entries 2 & 3 by Kimberly Pellechi [ July 25, 2003 ] 
3 
79027^{2} + 79029^{2} + ... + 79039^{2} + 79041^{2}  KP 
56.218.781.265  11 
2 
3465^{2} + 3467^{2} + ... + 3479^{2} + 3481^{2}  KP 
108.555.801  9 
1 
1^{2} + 3^{2} + 5^{2} + 7^{2} + 9^{2} + 11^{2} + 13^{2} + 15^{2} + 17^{2}  HS 
969  3 
  
Sums of Squares of Consecutive Odd Numbers 7 terms Contribution by Kimberly Pellechi [ July 1 & 25, 2003 ] 
2 
8567^{2} + 8569^{2} + 8571^{2} + 8573^{2} + 8575^{2} + 8577^{2} + 8579^{2}  KP 
514.474.415  9 
1 
5^{2} + 7^{2} + 9^{2} + 11^{2} + 13^{2} + 15^{2} + 17^{2}  KP 
959  3 
  
Sums of Squares of Consecutive Odd Numbers 5 terms Contribution by Kimberly Pellechi [ July 14, 2003 ] 
3 
10789^{2} + 10791^{2} + 10793^{2} + 10795^{2} + 10797^{2}  KP 
582.444.285  9 
2 
10395^{2} + 10397^{2} + 10399^{2} + 10401^{2} + 10403^{2}  KP 
540.696.045  9 
1 
331^{2} + 333^{2} + 335^{2} + 337^{2} + 339^{2}  KP 
561.165  6 
  
Sums of Squares of Consecutive Odd Numbers 3 terms Entries 3 & 4 by Kimberly Pellechi [ July 1, 2003 ] Extended by Patrick De Geest [ July 7, 2003 ] 
19 
71.818.189^{2} + 71.818.191^{2} + 71.818.193^{2}  PDG 
15.473.557.675.537.451  17 
18 
67.315.719^{2} + 67.315.721^{2} + 67.315.723^{2}  PDG 
13.594.218.881.249.531  17 
17 
41.386.135^{2} + 41.386.137^{2} + 41.386.139^{2}  PDG 
5.138.437.007.348.315  16 
16 
8.008.817^{2} + 8.008.819^{2} + 8.008.821^{2}  PDG 
192.423.545.324.291  15 
15 
7.181.807^{2} + 7.181.809^{2} + 7.181.811^{2}  PDG 
154.735.141.537.451  15 
14 
718.189^{2} + 718.191^{2} + 718.193^{2}  PDG 
1.547.394.937.451  13 
13 
413.861^{2} + 413.863^{2} + 413.865^{2}  PDG 
513.847.748.315  12 
12 
113.513^{2} + 113.515^{2} + 113.517^{2}  PDG 
38.656.965.683  11 
11 
11.373^{2} + 11.375^{2} + 11.377^{2}  PDG 
388.171.883  9 
10 
6.719^{2} + 6.721^{2} + 6.723^{2}  PDG 
135.515.531  9 
9 
4.135^{2} + 4.137^{2} + 4.139^{2}  PDG 
51.344.315  8 
8 
1.133^{2} + 1.135^{2} + 1.137^{2}  PDG 
3.864.683  7 
7 
759^{2} + 761^{2} + 763^{2}  PDG 
1.737.371  7 
6 
707^{2} + 709^{2} + 711^{2}  PDG 
1.508.051  7 
5 
197^{2} + 199^{2} + 201^{2}  PDG 
118.811  6 
4 
79^{2} + 81^{2} + 83^{2}  KP 
19.691  5 
3 
41^{2} + 43^{2} + 45^{2}  KP 
5.555  4 
2 
19^{2} + 21^{2} + 23^{2}  PDG 
1.331  4 
1 
11^{2} + 13^{2} + 15^{2}  PDG 
515  3 
  
Sums of Squares of Consecutive Even Numbers 3 terms Entries found by Patrick De Geest [ July 13, 2003 ] 
3 
16.403.468^{2} + 16.403.470^{2} + 16.403.472^{2}  PDG 
807.221.484.122.708  15 
2 
482.184^{2} + 482.186^{2} + 482.188^{2}  PDG 
697.510.015.796  12 
1 
145.424^{2} + 145.426^{2} + 145.428^{2}  PDG 
63.446.164.436  11 
  
Diophantine Equation  3^{rd} & 4^{th} powers Sources : Puzzle 47 & Puzzle 48
copied from Carlos Rivera's PP&P site. 
2 
69^{3} + 447^{3} + 893^{3}  3 terms 
929^{3}  3 
1 
30^{4} + 120^{4} + 272^{4} + 315^{4}  4 terms 
353^{4}  3 
  
Diophantine Equations  2^{nd} powers
From Hugo Sánchez [ May 3, 1999 ] 
3 
11011^{2} + 22022^{2} + 33033^{2} + 66066^{2} = 2 x 55055^{2}  4 terms 
2 x 55055^{2} = 33033^{2} + 44044^{2} + 55055^{2}  3 terms 
2 
212^{2} + 343^{2} + 424^{2} + 979^{2} = 1300810  4 terms 
1300810 = 555^{2} + 636^{2} + 767^{2}  3 terms 
1 
22^{2} + 33^{2} + 44^{2} + 99^{2} = 13310  4 terms 
13310 = 55^{2} + 66^{2} + 77^{2}  3 terms 
  
Palindromic Pattern from Sums of Consecutives By Carlos Rivera [ Feb 27, 1999 ] 
1 
S(2 + 3 + 4) = 9 S(2 + 3 + ... + 44) = 989 S(2 + 3 + ... + 444) = 98789 S(2 + 3 + ... + 4444) = 9876789 S(2 + 3 + ... + 44444) = 987656789 S(2 + 3 + ... + 444444) = 98765456789 S(2 + 3 + ... + 4444444) = 9876543456789 S(2 + 3 + ... + 44444444) = 987654323456789 S(2 + 3 + ... + 444444444) = 98765432123456789 S(2 + 3 + ... + 4444444444) = 9876543210123456789
 pattern is finite ! 
Thank you Carlos for this beautiful construction.  
  
Palindromic Pattern from Sums of Squared Palindromes By Hugo Sánchez [ May 3, 1999 ] 
1 
11^{2} + 22^{2} + ... + 66^{2} = 91 x 11^{2} = 11011
111^{2} + 222^{2} + ... + 666^{2} = 91 x 111^{2} = 1121211
1111^{2} + 2222^{2} + ... + 6666^{2} = 91 x 1111^{2} = 112323211
11111^{2} + 22222^{2} + ... + 66666^{2} = 91 x 11111^{2} = 11234343211
111111^{2} + 222222^{2} + ... + 666666^{2} = 91 x 111111^{2} = 1123454543211
1111111^{2} + 2222222^{2} + ... + 6666666^{2} = 91 x 1111111^{2} = 112345656543211
11111111^{2} + 22222222^{2} + ... + 66666666^{2} = 91 x 11111111^{2} = 11234567676543211
111111111^{2} + 222222222^{2} + ... + 666666666^{2} = 91 x 111111111^{2} = 1123456787876543211
1111111111^{2} + 2222222222^{2} + ... + 6666666666^{2} = 91 x 1111111111^{2} = 112345678989876543211
 pattern is finite ! 
Thanks Hugo for this beautiful pattern. Note that 91 is in fact a pseudopalindrome 1n1 See my palindromic squares page for more information  
  
Palindromic Sum of Powers from Consecutives By Carlos Rivera [ Feb 27, 1999 ] 
4 
1^{5} + 2^{5} + 3^{5} + 4^{5} + 5^{5} + 6^{5} + 7^{5} + 8^{5} + 9^{5} + 10^{5} + 11^{5} + 12^{5} + 13^{5}  13 terms 
1.002.001 1002001 = 1001^{2} = 7^{2} x 11^{2} x 13^{2}  7 
3 
1^{5} + 2^{5}  2 terms 
33  2 
2 
1^{4} + 2^{4} + 3^{4} + 4^{4} + 5^{4}  5 terms 
979  3 
1 
1^{2} + 2^{2} + 3^{2} +... ...+ 180^{2} + 181^{2} For sum of squares with max 5 terms see Sum of Squares.  181 terms 
1.992.991  7 
  
Palindromes from Consecutive Primes 2 to 23 and the Nine Digits Anagrams By definition the palindromes are always composite. By Carlos Rivera [ Feb 11, 1999 ] 
8 
2^{8} + 3^{9} + 5^{2} + 7^{4} + 11^{6} + 13^{1} + 17^{7} + 19^{3} + 23^{5}  9 terms 
418.575.814  9 
7 
2^{3} + 3^{6} + 5^{9} + 7^{5} + 11^{8} + 13^{7} + 17^{4} + 19^{2} + 23^{1}  9 terms 
279.161.972  9 
6 
2^{4} + 3^{7} + 5^{9} + 7^{6} + 11^{8} + 13^{5} + 17^{3} + 19^{1} + 23^{2}  9 terms 
216.808.612  9 
5 
2^{7} + 3^{8} + 5^{1} + 7^{9} + 11^{2} + 13^{4} + 17^{5} + 19^{6} + 23^{3}  9 terms 
88.866.888  8 
4 
2^{6} + 3^{9} + 5^{8} + 7^{4} + 11^{1} + 13^{7} + 17^{5} + 19^{3} + 23^{2}  9 terms 
64.588.546  8 
3 
2^{9} + 3^{8} + 5^{6} + 7^{3} + 11^{7} + 13^{2} + 17^{1} + 19^{4} + 23^{5}  9 terms 
26.077.062  8 
2 
2^{8} + 3^{7} + 5^{9} + 7^{6} + 11^{3} + 13^{4} + 17^{1} + 19^{5} + 23^{2}  9 terms 
4.579.754  7 
1 
2^{8} + 3^{9} + 5^{7} + 7^{1} + 11^{6} + 13^{4} + 17^{3} + 19^{5} + 23^{2}  9 terms 
4.379.734  7 
  
Palindromes from Consecutive Primes 3 to 29 and the Nine Digits Anagrams The palindromes have a chance to be prime. By Carlos Rivera [ Feb 11, 1999 ] 
4 
3^{9} + 5^{8} + 7^{4} + 11^{1} + 13^{3} + 17^{6} + 19^{2} + 23^{7} + 29^{5}  9 terms 
3.449.889.443  10 
3 
3^{7} + 5^{9} + 7^{8} + 11^{3} + 13^{4} + 17^{1} + 19^{5} + 23^{6} + 29^{2}  9 terms 
158.262.851  9 
2 
3^{4} + 5^{8} + 7^{9} + 11^{2} + 13^{7} + 17^{6} + 19^{5} + 23^{1} + 29^{3}  9 terms 
130.131.031  9 
1 
3^{9} + 5^{8} + 7^{7} + 11^{6} + 13^{5} + 17^{4} + 19^{3} + 23^{2} + 29^{1} Note that the primes and the 9digit anagram exponents are well ordered but in opposite direction ! See also WONplate 55  9 terms 
3.467.643  7 
  
Your contribution ? 
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