HOME plateWON | World!OfNumbers Various Palindromic Sums

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

Various Palindromic Sums

Index Nr Base Sequence Expression
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

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

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 !
9 terms
3.467.6437

1 ??
??

Contributions

Hugo Sánchez (email) a 'profesor de Educación Media que cultiva la Matemática Recreativa'
from Caracas, Venezuela found some interesting sequences- go to topic.

Carlos Rivera (email) found among others this beautiful pattern- go to topic.

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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