Various Palindromic Sums

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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
Index Nr	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
!	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 + 4228481²	KP
17	196.679.636.976.691	15
16	1011109² + 1011111² + ... + 1011127² + 1011129²	KP
16	11.245.977.954.211	14
15	424669² + 424671² + ... + 424687² + 424689²	KP
15	1.983.874.783.891	13
14	124951² + 124953² + ... + 124969² + 124971²	KP
14	171.767.767.171	12
13	119091² + 119093² + ... + 119109² + 119111²	KP
13	156.035.530.651	12
12	101109² + 101111² + ... + 101127² + 101129²	KP
12	112.475.574.211	12
11	42419² + 42421² + ... + 42437² + 42439²	KP
11	19.802.420.891	11
10	13161² + 13163² + ... + 13179² + 13181²	KP
10	1.908.228.091	10
9	11081² + 11083² + ... + 11099² + 11101²	KP
9	1.353.113.531	10
8	10121² + 10123² + ... + 10139² + 10141²	KP
8	1.129.009.211	10
7	10109² + 10111² + ... + 10127² + 10129²	KP
7	1.126.336.211	10
6	1191² + 1193² + 1195² + 1197² + 1199² + 1201² + 1203² + 1205² + 1207² + 1209² + 1211²	KP
6	15.866.851	8
5	1009² + 1011² + 1013² + 1015² + 1017² + 1019² + 1021² + 1023² + 1025² + 1027² + 1029²	KP
5	11.422.411	8
4	99² + 101² + 103² + 105² + 107² + 109² + 111² + 113² + 115² + 117² + 119²	KP
4	131.131	6
3	29² + 31² + 33² + 35² + 37² + 39² + 41² + 43² + 45² + 47² + 49²	KP
3	17.171	5
2	21² + 23² + 25² + 27² + 29² + 31² + 33² + 35² + 37² + 39² + 41²	HS
2	11.011	5
1	1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² + 17² + 19² + 21²	KP
1	1.771	4

	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 + 79041²	KP
3	56.218.781.265	11
2	3465² + 3467² + ... + 3479² + 3481²	KP
2	108.555.801	9
1	1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² + 17²	HS
1	969	3

	Sums of Squares of Consecutive Odd Numbers 7 terms Contribution by Kimberly Pellechi [July 1 & 25, 2003]
2	8567² + 8569² + 8571² + 8573² + 8575² + 8577² + 8579²	KP
2	514.474.415	9
1	5² + 7² + 9² + 11² + 13² + 15² + 17²	KP
1	959	3

	Sums of Squares of Consecutive Odd Numbers 5 terms Contribution by Kimberly Pellechi [July 14, 2003]
3	10789² + 10791² + 10793² + 10795² + 10797²	KP
3	582.444.285	9
2	10395² + 10397² + 10399² + 10401² + 10403²	KP
2	540.696.045	9
1	331² + 333² + 335² + 337² + 339²	KP
1	561.165	6

	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 + 71.818.193²	PDG
19	15.473.557.675.537.451	17
18	67.315.719² + 67.315.721² + 67.315.723²	PDG
18	13.594.218.881.249.531	17
17	41.386.135² + 41.386.137² + 41.386.139²	PDG
17	5.138.437.007.348.315	16
16	8.008.817² + 8.008.819² + 8.008.821²	PDG
16	192.423.545.324.291	15
15	7.181.807² + 7.181.809² + 7.181.811²	PDG
15	154.735.141.537.451	15
14	718.189² + 718.191² + 718.193²	PDG
14	1.547.394.937.451	13
13	413.861² + 413.863² + 413.865²	PDG
13	513.847.748.315	12
12	113.513² + 113.515² + 113.517²	PDG
12	38.656.965.683	11
11	11.373² + 11.375² + 11.377²	PDG
11	388.171.883	9
10	6.719² + 6.721² + 6.723²	PDG
10	135.515.531	9
9	4.135² + 4.137² + 4.139²	PDG
9	51.344.315	8
8	1.133² + 1.135² + 1.137²	PDG
8	3.864.683	7
7	759² + 761² + 763²	PDG
7	1.737.371	7
6	707² + 709² + 711²	PDG
6	1.508.051	7
5	197² + 199² + 201²	PDG
5	118.811	6
4	79² + 81² + 83²	KP
4	19.691	5
3	41² + 43² + 45²	KP
3	5.555	4
2	19² + 21² + 23²	PDG
2	1.331	4
1	11² + 13² + 15²	PDG
1	515	3

	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 + 16.403.472²	PDG
3	807.221.484.122.708	15
2	482.184² + 482.186² + 482.188²	PDG
2	697.510.015.796	12
1	145.424² + 145.426² + 145.428²	PDG
1	63.446.164.436	11

	Diophantine Equation - 3^rd & 4^th powers Sources : Puzzle 47 & Puzzle 48 copied from Carlos Rivera's PP&P site.
2	69³ + 447³ + 893³	3 terms
2	929³	3
1	30⁴ + 120⁴ + 272⁴ + 315⁴	4 terms
1	353⁴	3

	Diophantine Equations - 2^nd powers From Hugo Sánchez [May 3, 1999]
3	11011² + 22022² + 33033² + 66066² = 2 x 55055²	4 terms
3	2 x 55055² = 33033² + 44044² + 55055²	3 terms
2	212² + 343² + 424² + 979² = 1300810	4 terms
2	1300810 = 555² + 636² + 767²	3 terms
1	22² + 33² + 44² + 99² = 13310	4 terms
1	13310 = 55² + 66² + 77²	3 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 !
1	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 !
1	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
4	1.002.001 1002001 = 1001² = 7² x 11² x 13²	7
3	1⁵ + 2⁵	2 terms
3	33	2
2	1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴	5 terms
2	979	3
1	1² + 2² + 3² +... ...+ 180² + 181² For sum of squares with max 5 terms see Sum of Squares.	181 terms
1	1.992.991	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
8	418.575.814	9
7	2³ + 3⁶ + 5⁹ + 7⁵ + 11⁸ + 13⁷ + 17⁴ + 19² + 23¹	9 terms
7	279.161.972	9
6	2⁴ + 3⁷ + 5⁹ + 7⁶ + 11⁸ + 13⁵ + 17³ + 19¹ + 23²	9 terms
6	216.808.612	9
5	2⁷ + 3⁸ + 5¹ + 7⁹ + 11² + 13⁴ + 17⁵ + 19⁶ + 23³	9 terms
5	88.866.888	8
4	2⁶ + 3⁹ + 5⁸ + 7⁴ + 11¹ + 13⁷ + 17⁵ + 19³ + 23²	9 terms
4	64.588.546	8
3	2⁹ + 3⁸ + 5⁶ + 7³ + 11⁷ + 13² + 17¹ + 19⁴ + 23⁵	9 terms
3	26.077.062	8
2	2⁸ + 3⁷ + 5⁹ + 7⁶ + 11³ + 13⁴ + 17¹ + 19⁵ + 23²	9 terms
2	4.579.754	7
1	2⁸ + 3⁹ + 5⁷ + 7¹ + 11⁶ + 13⁴ + 17³ + 19⁵ + 23²	9 terms
1	4.379.734	7

	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
4	3.449.889.443	10
3	3⁷ + 5⁹ + 7⁸ + 11³ + 13⁴ + 17¹ + 19⁵ + 23⁶ + 29²	9 terms
3	158.262.851	9
2	3⁴ + 5⁸ + 7⁹ + 11² + 13⁷ + 17⁶ + 19⁵ + 23¹ + 29³	9 terms
2	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
1	3.467.643	7

	Your contribution ?
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Contributions

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

Carlos Rivera found among others this beautiful pattern- goto topic.

Kimberly Pellechi found many 'palindromic sums of the squares of the consecutive odd numbers'
or Pellechi Palindromes for short- goto topic.

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