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Various Palindromic Sums
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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 : 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 
   
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

flash
! 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
   
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
See my palindromic squares page for more information
 
   
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 !
See also WONplate 55
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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