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Palindromic Sums of
Squares of Consecutive Integers
rood Sums of Cubes rood Sums of Primes rood Sums of Powers rood
rood Various Palindromic Sums



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 : 535, 3773, 246191642

Sums of Squares of TWO Consecutive Integers
Sums of Squares of THREE Consecutive Integers
Sums of Squares of FOUR Consecutive Integers
Sums of Squares of FIVE Consecutive Integers


Palindromic Sums of Squares

A palindromic coincidence occurred with Five Consecutive Integers or is there a pattern ?

99 + 100 + 101 + 102 + 103 = 505
992 + 1002 + 1012 + 1022 + 1032 = 51015

10099 + 10100 + 10101 + 10102 + 10103 = 50505
100992 + 101002 + 101012 + 101022 + 101032 = 510151015

Two equations like 'number'-pregnancy. The children are the numbers 1981 and 699.
112 + 122 + 132 = 434
1198112 + 1198122 + 1198132 = 43064746034

372 + 382 + 392 = 4334
369972 + 369982 + 369992 = 4106556014




Messages

[ August 1, 2005 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, voici les numéros 39 et 40 qui hélas ne sont pas premiers (le numéro 39 a 5 facteurs premiers mais le numéro 40 est
un semi-prime -un "vrai"- avec un facteur de 15 chiffres et un autre de 17 chiffres ):

numéro 39 :
1682059335368470748635339502861 = 29 * 109 * 35935429 * 206697457 * 71640539617
9170766967294692 + 9170766967294702

numéro 40:
3167016920841776771480296107613 = 113665666219493 * 27862564186498841
12583753257358822 + 12583753257358832

Voilà c'est tout pour les palindromes de la forme 1... ...1 ou 3... ...3 de 31 chiffres. J'ai commencé à chercher ceux de la forme 5... ...5 .
Pour espérer trouver un palprime il faut donc s'attaquer aux nombres de 33 chiffres ( vraiment dur, dur ! ). A bientôt, JCR

[ September 6, 2005 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, voici le numéro 41 :

5450871426137276727316241780545 =
16508893703300162 + 16508893703300172

C'est le dernier des palindromes de 31 chiffres ! J'ai donc commencé la recherche des palindromes de
33 chiffres mais cela risque de prendre pas mal de temps... A bientôt. Kind regards. JCR

[ September 15, 2006 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, ca y est ! J'ai enfin trouvé le numéro 42. Le voici :

316370934175751979157571439073613 =
125771804108820822 + 125771804108820832

mais hélas il n'est pas premier (4 facteurs ) et comme j'ai fini l'étude des palindromes de la forme 31...13 pour trouver la cinquième
solution du puzzle 14 il faudra étudier les nombres de 35 chiffres... Je ne sais pas si j'aurai le temps et le courage !
Je vais quand même essayer de finir l'étude des palindromes de 33 chiffres de la forme 5.....5. JCR

[ October 14, 2006 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, ... et je viens de trouver le numéro 43. Le voici:

501263304966749757947669403362105 =
158313503051184762 + 158313503051184772

Voilà, c'est tout pour aujourd'hui... J'espère arriver un jour au numéro 50 ;-) JCR

[ May 21, 2007 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, Ca y est ! J'ai enfin trouvé le numéro 44. Le voici :

562318868215014101410512868813265 =
167678094606155112 + 167678094606155122

Amitiés. JCR

[ January 12, 2008 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, ... avec un peu de retard voici le numéro 45 :

588186547187469040964781745681885 =
171491478970161812 + 171491478970161822

Best regards. JCR

[ February 17, 2009 ]
Jean-Claude Rosa (email)
[ goto entry ]
Cher Patrick, Voilà dejà un an que je n'ai pas eu de tes nouvelles ( le temps passe... ) mais j'attendais de trouver le numéro 46
pour t'écrire. Ca y est ! mais hélas il n'est pas premier (petite consolation c'est un semi-prime !). Le voici :

10489844562990321812309926544898401 = 2344709 * 4473836438974014179290447789
724218356678092002 + 724218356678092012

A bientôt. JCR




Sources Revealed

Huen Y.K. from Singapore developed a general generating function for palindromic sums and products
of consecutive integers using concise programcode written for Macsyma 2.2.1.
Global Generating Function For Palindromic Sums and Products of Consecutive Integers.

See also Powers of Consecutives Summing to Palindromes and WONplate 145



Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
separator
The regular numbers of form n2 + (n+1)2 [sums of two squared consecutives] :
%N Centered square numbers: 2n(n-1)+1. under A001844.
The subsets in relation to the palindromic and prime numbers of above form :
%N n^2 + (n+1)^2 is palindromic. under A027571.
%N Palindromes of form n^2 + (n+1)^2. under A027572.
%N n^2 + (n+1)^2 is prime. under A027861.
%N Primes of form n^2 + (n+1)^2. under A027862.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 [sums of three squared consecutives] :
%N Points on surface of square pyramid: 3*n^2 + 2. under A005918.
The subsets in relation to the palindromic and prime numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 is palindromic. under A027573.
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2. under A027574.
%N n^2 + (n+1)^2 + (n+2)^2 is prime. under A027863.
%N Primes of form n^2 + (n+1)^2 + (n+2)^2. under A027864.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 [sums of four squared consecutives] :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2. under A027575.
The subsets in relation to the palindromic numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 is palindromic. under A027576.
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2. under A027577.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 + (n+4)2 [sums of five squared consecutives] :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2. under A027578.
The subsets in relation to the palindromic numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 is palindromic. under A027579.
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2. under A027580.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 + (n+4)2 + (n+5)2 :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2. under A027865.
The subsets in relation to the prime numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2 is prime. under A027866.
%N Primes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2. under A027867.
separator
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.


A simple substitution of n with (m–1) will proof that n^2 + (n+1)^2 + (n+2)^2 equals 3*n^2 + 2 :

n^2 + (n+1)^2 + (n+2)^2 =
(m–1)^2 + ((m–1)+1)^2 + ((m–1)+2)^2 =
(m^2–2m+1) + (m^2) + (m^2+2m+1) =
m^2 + 1 + m^2 + m^2 + 1 =
3*m^2 + 2 QED
And also that n^2 + (n+1)^2 equals 2n(n–1)+1 :
n^2 + (n+1)^2 =
(m–1)^2 + ((m–1)+1)^2 =
m^2–2m+1 + m^2 =
2m^2 – 2m + 1 =
2m(m–1) + 1 QED







The Table



Index Nr Base Square Expression 
Palindromic Sums of Squares of Consecutive Integers Length 
   
Sums of Squares of FIVE Consecutive Integers
[PDG] Searched upto length 17.
Entry from Chai Wah Wu (OEIS)
Index 6 [ January 18, 2016 ]
Entry from Giovanni Resta (OEIS)
Index 7 [ August 04, 2019 ]
7 10.310.949.321.2532 + 10.310.949.321.2542 + 10.310.949.321.2552 +
10.310.949.321.2562 + 10.310.949.321.2572
14
531.578.379.527.444.725.973.875.13527
6 105.720.053.6352 + 105.720.053.6362 + 105.720.053.6372 +
105.720.053.6382 + 105.720.053.6392
12
55.883.648.705.050.784.638.85523
5 10.0992 + 10.1002 + 10.1012 + 10.1022 + 10.10325
510.151.0159
4 3.2072 + 3.2082 + 3.2092 + 3.2102 + 3.21124
51.488.4158
3 3312 + 3322 + 3332 + 3342 + 33523
554.4556
2 992 + 1002 + 1012 + 1022 + 10322 - 3
51.0155
1 12 + 22 + 32 + 42 + 521
552
   
Sums of Squares of FOUR Consecutive Integers
(Palindromes are of odd-length only)
[PDG] Searched upto length 17.
Next entries from Donovan Johnson (OEIS)
Index 7 & 8 [ August 26, 2012 ]
8 1.086.219.2422 + 1.086.219.2432 + 1.086.219.2442 + 1.086.219.245210
4.719.488.979.798.849.17419
7 1.085.864.4922 + 1.085.864.4932 + 1.085.864.4942 + 1.085.864.495210
4.716.406.792.976.046.17419
6 102.025.3002 + 102.025.3012 + 102.025.3022 + 102.025.30329
41.636.648.584.663.61417
5 101.740.1722 + 101.740.1732 + 101.740.1742 + 101.740.17529
41.404.251.615.240.41417
4 12.309.5982 + 12.309.5992 + 12.309.6002 + 12.309.60128
606.104.959.401.60615
3 10.460.1372 + 10.460.1382 + 10.460.1392 + 10.460.14028
437.657.989.756.73415
2 10.1722 + 10.1732 + 10.1742 + 10.17525
414.000.4149
1 1002 + 1012 + 1022 + 10323
41.2145
   
Sums of Squares of THREE Consecutive Integers
Submissions from Jean Claude Rosa (email)
Searched for palindromes upto length 21.
Index 22 and 23 [ August 28, 2002 ]
Index 24 up to 28 [ August 30, 2002 ]
28 17.552.348.1962 + 17.552.348.1972 + 17.552.348.198211
924.254.781.686.187.452.42921
27 16.028.474.3442 + 16.028.474.3452 + 16.028.474.346211
770.735.969.484.969.537.07721
26 12.900.321.3912 + 12.900.321.3922 + 12.900.321.393211
499.254.876.050.678.452.99421
25 8.227.749.4892 + 8.227.749.4902 + 8.227.749.491210
203.087.585.010.585.780.30221
24 5.602.945.2422 + 5.602.945.2432 + 5.602.945.244210
94.178.986.188.168.987.14920
23 828.223.2492 + 828.223.2502 + 828.223.25129
2.057.861.255.521.687.50219
22 441.564.3802 + 441.564.3812 + 441.564.38229
584.937.307.703.739.48518
21 40.439.5572 + 40.439.5582 + 40.439.55928
4.906.073.553.706.09416
20 17.909.2822 + 17.909.2832 + 17.909.28428
962.227.252.722.26915
19 8.211.8792 + 8.211.8802 + 8.211.88127
202.304.919.403.20215
18 4.427.7802 + 4.427.7812 + 4.427.78227
58.815.733.751.88514
17 1.775.7062 + 1.775.7072 + 1.775.70827
9.459.406.049.54913
16 1.751.4962 + 1.751.4972 + 1.751.49827
Prime Curios!    9.203.225.223.02913
15 1.328.1202 + 1.328.1212 + 1.328.12227
5.291.716.171.92513
14 378.8112 + 378.8122 + 378.81326
430.495.594.03412
13 173.7362 + 173.7372 + 173.73826
90.553.635.50911
12 119.8112 + 119.8122 + 119.81326
43.064.746.03411
11 40.4412 + 40.4422 + 40.44325
4.906.666.09410
10 36.9972 + 36.9982 + 36.99925
4.106.556.01410
9 17.9322 + 17.9332 + 17.93425
964.777.4699
8 17.5522 + 17.5532 + 17.55425
924.323.4299
7 16.0842 + 16.0852 + 16.08625
776.181.6779
6 8.3392 + 8.3402 + 8.34124
208.666.8029
5 1.7362 + 1.7372 + 1.73824
9.051.5097
4 5662 + 5672 + 56823
964.4696
3 372 + 382 + 3922
4.3344
2 112 + 122 + 1322
4343
1 42 + 52 + 621
772
   
Sums of Squares of TWO Consecutive Integers
Also Palindromic Centered Squares
(See my page Palindromic Centered Polygonals)

Palindromes are of odd length and nature only.
Carlos Rivera collects these palindromes (Palprimes and sums of powers)
especially when they are prime (see highlights).
Submissions from Jean Claude Rosa (email)
Index 27 [ September 14, 2002 ]
Index 28 [ October 11, 2002 ]
Index 29, 30, 31 & 32 [ October 17, 18, 19 & 21, 2002 ]
Index 33 [ February 10, 2003 ]
Index 34 & 35 [ February 26, 2003 ]
Index 36 & 37 [ June 8, 2005 ]
Index 38 [ July 5, 2005 ]
Index 39 & 40 [ August 1, 2005 ]
Index 41 [ September 6, 2005 ]
Index 42 [ September 15, 2006 ]
Index 43 [ October 14, 2006 ]
Index 44 [ May 21, 2007 ]
Index 45 [ January 12, 2008 ]
Index 46 [ February 17, 2009 ]
Search resumed by Patrick De Geest
Index 47 to 49 [ May 2, 2021 ]
49 124.566.258.570.164.6972 + 124.566.258.570.164.698218
31.033.505.548.338.259.895.283.384.550.533.01335
48 91.424.589.460.359.8552 + 91.424.589.460.359.856217
16.716.911.115.990.684.748.609.951.111.961.76135
47 89.663.751.709.213.5052 + 89.663.751.709.213.506217
16.079.176.741.142.975.657.924.114.767.197.06135
46 72.421.835.667.809.2002 + 72.421.835.667.809.201217
10.489.844.562.990.321.812.309.926.544.898.40135
45 17.149.147.897.016.1812 + 17.149.147.897.016.182217
588.186.547.187.469.040.964.781.745.681.88533
44 16.767.809.460.615.5112 + 16.767.809.460.615.512217
562.318.868.215.014.101.410.512.868.813.26533
43 15.831.350.305.118.4762 + 15.831.350.305.118.477217
501.263.304.966.749.757.947.669.403.362.10533
42 12.577.180.410.882.0822 + 12.577.180.410.882.083217
316.370.934.175.751.979.157.571.439.073.61333
41 1.650.889.370.330.0162 + 1.650.889.370.330.017216
5.450.871.426.137.276.727.316.241.780.54531
40 1.258.375.325.735.8822 + 1.258.375.325.735.883216
3.167.016.920.841.776.771.480.296.107.61331
39 917.076.696.729.4692 + 917.076.696.729.470215
1.682.059.335.368.470.748.635.339.502.86131
38 712.254.882.551.4252 + 712.254.882.551.426215
1.014.614.035.436.689.866.345.304.164.10131
37 86.505.526.088.5702 + 86.505.526.088.571214
14.966.412.087.720.702.778.021.466.94129
36 73.675.999.227.0992 + 73.675.999.227.100214
10.856.305.724.223.132.242.750.365.80129
35 12.615.243.893.5622 + 12.615.243.893.563214
318.288.756.988.131.889.657.882.81327
34 12.613.810.689.7622 + 12.613.810.689.763214
318.216.440.234.333.432.044.612.81327
33 7.365.154.696.7752 + 7.365.154.696.776213
108.491.007.414.868.414.700.194.80127
32 1.589.811.006.1232 + 1.589.811.006.124213
5.054.998.070.382.830.708.994.50525
31 1.258.837.943.7172 + 1.258.837.943.718213
3.169.345.937.085.807.395.439.61325
30 1.255.589.965.8522 + 1.255.589.965.853213
3.153.012.324.698.964.232.103.51325
29 1.250.999.800.0122 + 1.250.999.800.013213
3.130.000.999.262.629.990.000.31325
28 844.706.005.2202 + 844.706.005.221212
1.427.056.470.511.150.746.507.24125
27 165.058.650.6662 + 165.058.650.667212
54.488.716.319.691.361.788.44523
26 16.794.058.7112 + 16.794.058.712211
564.080.816.010.618.080.46521
25 12.574.461.6172 + 12.574.461.618211
316.234.169.939.961.432.61321
24 8.055.329.3602 + 8.055.329.361210
129.776.662.212.266.677.92121
23 8.032.814.1392 + 8.032.814.140210
129.052.205.999.502.250.92121
22 7.360.311.9002 + 7.360.311.901210
108.348.382.545.283.843.80121
21 1.705.387.6432 + 1.705.387.644210
5.816.694.029.204.966.18519
20 1.680.689.7882 + 1.680.689.789210
5.649.436.330.336.349.46519
19 1.616.689.8032 + 1.616.689.804210
5.227.371.841.481.737.22519
18 80.472.2642 + 80.472.26528
12.951.570.707.515.92117
17 17.070.7062 + 17.070.70728
582.818.040.818.28515
16 12.458.5972 + 12.458.59828
310.433.303.334.01315
15 8.659.9292 + 8.659.93027
149.988.757.889.94115
14 7.901.0142 + 7.901.01527
124.852.060.258.42115
13 1.258.1822 + 1.258.18327
3.166.046.406.61313
12 904.2802 + 904.28126
1.635.446.445.36113
11 170.7062 + 170.70726
58.281.418.28511
10 1.7062 + 1.70724
5.824.2857
9 1.6212 + 1.62224
5.258.5257
8 1.2622 + 1.26324
Prime Curios!    3.187.8137
7 1.2572 + 1.25824
3.162.6137
6 9192 + 92023
1.690.9617
5 162 + 1722
5453
4 122 + 1322
Prime!    3133
3 92 + 1021 - 2
Prime!    1813
2 12 + 221
Prime!    51
1 02 + 121
11


Contributions

Jean-Claude Rosa - go to entry sum of 3 squares - go to entry sum of 2 squares




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E-mail address : pdg@worldofnumbers.com