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Various Palindromic Sums
rood Sums of Squares rood Sums of Cubes rood Sums of Primes rood Sums of Powers rood
rood Sum of First Numbertypes rood Sequence Products rood Reversal Products rood Pythagorean Triples rood
rood Palindromes in other Bases rood Palindromes in Concatenations


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

  1. Multigrade Palprimes Patterns
  2. Sums of Squares of Consecutive Odd Numbers (11 terms)
  3. Sums of Squares of Consecutive Odd Numbers (9 terms)
  4. Sums of Squares of Consecutive Odd Numbers (7 terms)
  5. Sums of Squares of Consecutive Odd Numbers (5 terms)
  6. Sums of Squares of Consecutive Odd Numbers (3 terms)
  7. Sums of Squares of Consecutive Even Numbers (3 terms)
  8. Diophantine Equation - 3rd & 4th powers
  9. Diophantine Equations - 2nd powers
  10. Palindromic Pattern from Sums of Consecutives
  11. Palindromic Pattern from Sums of Squared Palindromes
  12. Palindromic Sum of Powers from Consecutives
  13. Palindromes from Consecutive Primes 2 to 23 and the Nine Digits Anagrams
  14. Palindromes from Consecutive Primes 3 to 29 and the Nine Digits Anagrams
  15. Your contribution ?


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
   
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 42284612 + 42284632 + ... + 42284792 + 42284812KP
196.679.636.976.69115
16 10111092 + 10111112 + ... + 10111272 + 10111292KP
11.245.977.954.21114
15 4246692 + 4246712 + ... + 4246872 + 4246892KP
1.983.874.783.89113
14 1249512 + 1249532 + ... + 1249692 + 1249712KP
171.767.767.17112
13 1190912 + 1190932 + ... + 1191092 + 1191112KP
156.035.530.65112
12 1011092 + 1011112 + ... + 1011272 + 1011292KP
112.475.574.21112
11 424192 + 424212 + ... + 424372 + 424392KP
19.802.420.89111
10 131612 + 131632 + ... + 131792 + 131812KP
1.908.228.09110
9 110812 + 110832 + ... + 110992 + 111012KP
1.353.113.53110
8 101212 + 101232 + ... + 101392 + 101412KP
1.129.009.21110
7 101092 + 101112 + ... + 101272 + 101292KP
1.126.336.21110
6 11912 + 11932 + 11952 + 11972 + 11992 + 12012 +
12032 + 12052 + 12072 + 12092 + 12112		KP
15.866.8518
5 10092 + 10112 + 10132 + 10152 + 10172 + 10192 +
10212 + 10232 + 10252 + 10272 + 10292		KP
11.422.4118
4 992 + 1012 + 1032 + 1052 + 1072 + 1092 +
1112 + 1132 + 1152 + 1172 + 1192		KP
131.1316
3 292 + 312 + 332 + 352 + 372 + 392 +
412 + 432 + 452 + 472 + 492		KP
17.1715
2 212 + 232 + 252 + 272 + 292 + 312 +
332 + 352 + 372 + 392 + 412		HS
11.0115
1 12 + 32 + 52 + 72 + 92 + 112 +
132 + 152 + 172 + 192 + 212		KP
1.7714
   
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 790272 + 790292 + ... + 790392 + 790412KP
56.218.781.26511
2 34652 + 34672 + ... + 34792 + 34812KP
108.555.8019
1 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172HS
9693
   
Sums of Squares of Consecutive Odd Numbers
7 terms
Contribution by Kimberly Pellechi [ July 1 & 25, 2003 ]
2 85672 + 85692 + 85712 + 85732 + 85752 + 85772 + 85792KP
514.474.4159
1 52 + 72 + 92 + 112 + 132 + 152 + 172KP
9593
   
Sums of Squares of Consecutive Odd Numbers
5 terms
Contribution by Kimberly Pellechi [ July 14, 2003 ]
3 107892 + 107912 + 107932 + 107952 + 107972KP
582.444.2859
2 103952 + 103972 + 103992 + 104012 + 104032KP
540.696.0459
1 3312 + 3332 + 3352 + 3372 + 3392KP
561.1656
   
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 71.818.1892 + 71.818.1912 + 71.818.1932PDG
15.473.557.675.537.45117
18 67.315.7192 + 67.315.7212 + 67.315.7232PDG
13.594.218.881.249.53117
17 41.386.1352 + 41.386.1372 + 41.386.1392PDG
5.138.437.007.348.31516
16 8.008.8172 + 8.008.8192 + 8.008.8212PDG
192.423.545.324.29115
15 7.181.8072 + 7.181.8092 + 7.181.8112PDG
154.735.141.537.45115
14 718.1892 + 718.1912 + 718.1932PDG
1.547.394.937.45113
13 413.8612 + 413.8632 + 413.8652PDG
513.847.748.31512
12 113.5132 + 113.5152 + 113.5172PDG
38.656.965.68311
11 11.3732 + 11.3752 + 11.3772PDG
388.171.8839
10 6.7192 + 6.7212 + 6.7232PDG
135.515.5319
9 4.1352 + 4.1372 + 4.1392PDG
51.344.3158
8 1.1332 + 1.1352 + 1.1372PDG
3.864.6837
7 7592 + 7612 + 7632PDG
1.737.3717
6 7072 + 7092 + 7112PDG
1.508.0517
5 1972 + 1992 + 2012PDG
118.8116
4 792 + 812 + 832KP
19.6915
3 412 + 432 + 452KP
5.5554
2 192 + 212 + 232PDG
1.3314
1 112 + 132 + 152PDG
5153
   
Sums of Squares of Consecutive Even Numbers
3 terms
Entries found by Patrick De Geest [ July 13, 2003 ]
3 16.403.4682 + 16.403.4702 + 16.403.4722PDG
807.221.484.122.70815
2 482.1842 + 482.1862 + 482.1882PDG
697.510.015.79612
1 145.4242 + 145.4262 + 145.4282PDG
63.446.164.43611
   
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 110112 + 220222 + 330332 + 660662 = 2 x 5505524 terms
2 x 550552 = 330332 + 440442 + 5505523 terms
2 2122 + 3432 + 4242 + 9792 = 13008104 terms
1300810 = 5552 + 6362 + 76723 terms
1 222 + 332 + 442 + 992 = 133104 terms
13310 = 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 + 135		13 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 with max 5 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 9 and topic 11.

Carlos Rivera (email) found among others this beautiful pattern- go to topic 10, topic 12, topic 13 and topic 14.

Kimberly Pellechi (email) found many 'palindromic sums of the squares of the consecutive odd numbers'
or Pellechi Palindromes for short- go to topic.
 

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E-mail address : pdg@worldofnumbers.com