HOME plateWON | World!OfNumbers Various Palindromic Sums Sums of Squares Sums of Cubes Sums of Primes Sums of Powers Sum of First Numbertypes Sequence Products Reversal Products Pythagorean Triples Palindromes in other Bases Palindromes in Concatenations

Introduction

Palindromic numbers are numbers which read the same from
left to right (forwards) as from the right to left (backwards)
Here are a few random examples : 353, 37173, 24611642

 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

Various Palindromic Sums

Index Nr Base Sequence ExpressionInitials
Various Palindromic Sums Length

Contribution by Carlos Rivera [ August 13, 1999 ]
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

from
A. Beiler's
book
page 164

Diophantine Equation - 3rd & 4th powers
Sources : Puzzle 47 & Puzzle 48
copied from Carlos Rivera's PP&P site.
2 693 + 4473 + 89333 terms
92933
1 304 + 1204 + 2724 + 31544 terms
35343

Diophantine Equations - 2nd powers
From Hugo Sánchez [ May 3, 1999 ]
3 11.0112 + 22.0222 + 33.0332 + 66.0662 = 2 x 55.05524 terms
2 x 55.0552 = 33.0332 + 44.0442 + 55.05523 terms
2 2122 + 3432 + 4242 + 9792 = 1.300.8104 terms
1.300.810 = 5552 + 6362 + 76723 terms
1 222 + 332 + 442 + 992 = 13.3104 terms
13.310 = 552 + 662 + 7723 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 ]
1 112 + 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

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 + 13513 terms
1.002.001
1002001 = 10012 = 72 x 112 x 132
7
3 15 + 252 terms
332
2 14 + 24 + 34 + 44 + 545 terms
9793
1 12 + 22 + 32 +... ...+ 1802 + 1812
For sum of squares up to 11 terms see Sum of Squares.
181 terms
1.992.9917

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 + 2359 terms
418.575.8149
7 23 + 36 + 59 + 75 + 118 + 137 + 174 + 192 + 2319 terms
279.161.9729
6 24 + 37 + 59 + 76 + 118 + 135 + 173 + 191 + 2329 terms
216.808.6129
5 27 + 38 + 51 + 79 + 112 + 134 + 175 + 196 + 2339 terms
88.866.8888
4 26 + 39 + 58 + 74 + 111 + 137 + 175 + 193 + 2329 terms
64.588.5468
3 29 + 38 + 56 + 73 + 117 + 132 + 171 + 194 + 2359 terms
26.077.0628
2 28 + 37 + 59 + 76 + 113 + 134 + 171 + 195 + 2329 terms
4.579.7547
1 28 + 39 + 57 + 71 + 116 + 134 + 173 + 195 + 2329 terms
4.379.7347

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 + 2959 terms
3.449.889.44310
3 37 + 59 + 78 + 113 + 134 + 171 + 195 + 236 + 2929 terms
158.262.8519
2 34 + 58 + 79 + 112 + 137 + 176 + 195 + 231 + 2939 terms
130.131.0319
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 !
9 terms
3.467.6437

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 + 6189 terms
218.175.385.351.779 (base 10)
110001100110110111101010010101111011011001100011 (base 2)
15
48

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 + 14439 terms
6.490.660.94610

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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