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 Various Palindromic Sums | |||
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Introduction
left to right (forwards)
as from the right to left (backwards) 
| Multigrade Palprimes Patterns | |
| Diophantine Equation - 3rd & 4th powers | |
| Diophantine Equations - 2nd powers | |
| Palindromic Pattern from Sums of Consecutives | |
| Palindromic Pattern from Sums of Squared Palindromes | |
| Palindromic Sum of Powers from Consecutives | |
| Palindromes from Consecutive Primes 2 to 23 and the Nine Digits Anagrams | |
| Palindromes from Consecutive Primes 3 to 29 and the Nine Digits Anagrams | |
| A record palindrome using startprime 29, ninedigital 792436518 and base 2 | |
| A record palindrome using consecutive Fibonacci terms and ninedigital 947862153 | |
| 44 palindromes arising from the sum of ten consecutive Fibonacci terms to the power of the digits of pandigitals |
Various Palindromic Sums
| Index Nr | Base Sequence Expression | Initials |
|---|---|---|
| Various Palindromic Sums | Length | |
| [VARIA 1] 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 105012 + 147412 + 154512 = 114112 + 127212 + 165612 |
|
| 1 | 181 + 727 + 757 = 353 + 383 + 929 1812 + 7272 + 7572 = 3532 + 3832 + 9292 |
 |
| ! | 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 130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 22066126024 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972 130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 1642056213257460 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973 |
Trigrade from A. Beiler's book page 164 |
| [VARIA 2] Diophantine Equation - 3rd & 4th powers Sources : Puzzle 47 & Puzzle 48 copied from Carlos Rivera's PP&P site. | ||
| 2 | 693 + 4473 + 8933 | 3 terms |
| 9293 | 3 | |
| 1 | 304 + 1204 + 2724 + 3154 | 4 terms |
| 3534 | 3 | |
| [VARIA 3] Diophantine Equations - 2nd powers From Hugo Sánchez [ May 3, 1999 ] | ||
| 3 | 11.0112 + 22.0222 + 33.0332 + 66.0662 = 2 x 55.0552 | 4 terms |
| 2 x 55.0552 = 33.0332 + 44.0442 + 55.0552 | 3 terms | |
| 2 | 2122 + 3432 + 4242 + 9792 = 1.300.810 | 4 terms |
| 1.300.810 = 5552 + 6362 + 7672 | 3 terms | |
| 1 | 222 + 332 + 442 + 992 = 13.310 | 4 terms |
| 13.310 = 552 + 662 + 772 | 3 terms | |
| [VARIA 4] 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. | ||
| [VARIA 5] Palindromic Pattern from Sums of Squared Palindromes By Hugo Sánchez [ May 3, 1999 ] | ||
| 1 | 112 + 222 + ... + 662 = 91 x 112 = 11011 1112 + 2222 + ... + 6662 = 91 x 1112 = 1121211 11112 + 22222 + ... + 66662 = 91 x 11112 = 112323211 111112 + 222222 + ... + 666662 = 91 x 111112 = 11234343211 1111112 + 2222222 + ... + 6666662 = 91 x 1111112 = 1123454543211 11111112 + 22222222 + ... + 66666662 = 91 x 11111112 = 112345656543211 111111112 + 222222222 + ... + 666666662 = 91 x 111111112 = 11234567676543211 1111111112 + 2222222222 + ... + 6666666662 = 91 x 1111111112 = 1123456787876543211 11111111112 + 22222222222 + ... + 66666666662 = 91 x 11111111112 = 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 | ||
| [VARIA 6] Palindromic Sum of Powers from Consecutives By Carlos Rivera [ Feb 27, 1999 ] | ||
| 4 | 15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95 + 105 + 115 + 125 + 135 | 13 terms |
| 1.002.001 1002001 = 10012 = 72 x 112 x 132 | 7 | |
| 3 | 15 + 25 | 2 terms |
| 33 | 2 | |
| 2 | 14 + 24 + 34 + 44 + 54 | 5 terms |
| 979 | 3 | |
| 1 | 12 + 22 + 32 +... ...+ 1802 + 1812 For sum of squares up to 11 terms see Sum of Squares. | 181 terms |
| 1.992.991 | 7 | |
| [VARIA 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 | 28 + 39 + 52 + 74 + 116 + 131 + 177 + 193 + 235 | 9 terms |
| 418.575.814 | 9 | |
| 7 | 23 + 36 + 59 + 75 + 118 + 137 + 174 + 192 + 231 | 9 terms |
| 279.161.972 | 9 | |
| 6 | 24 + 37 + 59 + 76 + 118 + 135 + 173 + 191 + 232 | 9 terms |
| 216.808.612 | 9 | |
| 5 | 27 + 38 + 51 + 79 + 112 + 134 + 175 + 196 + 233 | 9 terms |
| 88.866.888 | 8 | |
| 4 | 26 + 39 + 58 + 74 + 111 + 137 + 175 + 193 + 232 | 9 terms |
| 64.588.546 | 8 | |
| 3 | 29 + 38 + 56 + 73 + 117 + 132 + 171 + 194 + 235 | 9 terms |
| 26.077.062 | 8 | |
| 2 | 28 + 37 + 59 + 76 + 113 + 134 + 171 + 195 + 232 | 9 terms |
| 4.579.754 | 7 | |
| 1 | 28 + 39 + 57 + 71 + 116 + 134 + 173 + 195 + 232 | 9 terms |
| 4.379.734 | 7 | |
| [VARIA 8] 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 | 39 + 58 + 74 + 111 + 133 + 176 + 192 + 237 + 295 | 9 terms |
| 3.449.889.443 | 10 | |
| 3 | 37 + 59 + 78 + 113 + 134 + 171 + 195 + 236 + 292 | 9 terms |
| 158.262.851 | 9 | |
| 2 | 34 + 58 + 79 + 112 + 137 + 176 + 195 + 231 + 293 | 9 terms |
| 130.131.031 | 9 | |
| 1 | 39 + 58 + 77 + 116 + 135 + 174 + 193 + 232 + 291 Note that the primes and the 9-digit anagram exponents are well ordered but in opposite direction ! See also WONplate 55 | 9 terms |
| 3.467.643 | 7 | |
| [VARIA 9] A record palindrome using startprime 29, ninedigital 792436518 and base 2 By PDG [ Jun 10, 2022 ] | ||
| 1 | 297 + 319 + 372 + 414 + 433 + 476 + 535 + 591 + 618 | 9 terms |
| 218.175.385.351.779 (base 10) 110001100110110111101010010101111011011001100011 (base 2) | 15 48 | |
| [VARIA 10] A record palindrome using consecutive Fibonacci terms and ninedigital 947862153 By Alexandru Petrescu [ Jun 17, 2022 ] | ||
| 1 | 39 + 54 + 87 + 138 + 216 + 342 + 551 + 895 + 1443 | 9 terms |
| 6.490.660.946 | 10 | |
| [VARIA 11] 44 palindromes arising from the sum of ten consecutive Fibonacci terms to the power of the digits of pandigitals By Alexandru Petrescu [ Nov 26, 2025 ] |
For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.
Varia 7 → 892461735, 369587421, 479685312, 781924563, 698417532, 986372145, 879634152, 897164352
Varia 8 → 984136275, 798341562, 489276513, 987654321
Varia 11 → 2548319076, 8394126750, 3189460752, 6872391405, 9380172645, 7893421605, 5819407623, 8907325641, 3769582041, 7069583412, 8067943152, 2798160354, 9780462351, 6854973120, 5786932140, 7386945201, 2678951430, 7658920431, 2694871503, 2986573104, 8962170543, 8971603524, 7495816203, 3974856201, 8739562014, 6829715430, 9378465210, 678905432, 87594662103, 8769235410, 8950764321, 7869145023, 4986735012, 8659720431, 6589732401, 1689754032, 7984615032, 8579632401, 9680754213, 9286754031, 7986352401, 7986503412, 7598634021, 9876543120
Contributions
Hugo Sánchez (email) a 'profesor de Educación Media que cultiva la Matemática Recreativa'
from Caracas, Venezuela found some interesting sequences- go to topic 3 and topic 5.
Carlos Rivera (email) found among others these beautiful patterns- go to topic 2, go to topic 4, topic 6, topic 7 and topic 8.
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Patrick De Geest - Belgium
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