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WON |
Full List of the
Sporadic Square Palindromes (SSP)
rood Palindromic Squares rood All Squares up to length 31 rood
rood The Subsets rood Factorization


Legend

In the table below you'll find all the Sporadic Square Palindromes (SSP's) I collected so far.
Not included in the list are the Binary, Ternary, Even Root and Asymmetric family members.

 sidecells background color designate the sporadic square palindromes with EVEN length
 sidecells background color designate the sporadic square palindromes with ODD length

 PLAIN TEXT SQUARES      PLAIN TEXT SSP 

So, numbers with a palindromic basenumber are not listed 'exhaustively' as they become too numerous.
To keep the aesthetic look of the table acceptable only the most interesting palindromic numbers are
restrained i.e. the sporadics or those with a non-palindromic basenumber not part of any infinite family.


The Table


Index NrInfoBasenumberLength
Sporadic Square Palindromic NumberLength
   
[PG4] Formula = n^2
See also A059744 and A059745
p4Info 31.346.828.297.209.660.045.268.842.120.992.23335
982.623.644.294.744.275.088.611.239.676.071.787.170.676.932.116.880.572.447.492.446.326.28969
p3Info 25.980.813.040.115.834.424.688.125.458.775.02435
675.002.646.225.452.986.662.849.713.318.939.080.939.813.317.948.266.689.254.522.646.200.57669
p2Info 25.584.361.094.815.754.876.257.141.366.406.38435
654.559.532.629.922.011.473.373.439.846.828.282.828.648.934.373.374.110.229.926.235.955.45669
p1Info 21.471.116.183.870.468.837.446.052.903.107.64235
461.008.830.181.264.364.574.728.001.751.741.929.147.157.100.827.475.463.462.181.038.800.16469
8729Info 3.109.885.844.380.152.888.763.910.885.950.36334
9.671.389.965.076.056.510.789.702.463.424.611.164.243.642.079.870.156.506.705.699.831.76967
8728Info 2.602.189.663.685.835.935.193.356.696.699.42434
6.771.391.045.793.403.931.618.552.334.848.680.868.484.332.558.161.393.043.975.401.931.77667
8727Info 2.199.198.533.628.007.527.429.007.800.357.37234
4.836.474.190.311.578.555.464.055.886.890.474.740.986.885.504.645.558.751.130.914.746.38467
8028Info 982.503.990.036.767.976.718.486.749.830.91333
965.314.090.438.169.467.661.632.561.053.978.879.350.165.236.166.764.961.834.090.413.56966
8027Info 633.801.296.960.665.546.843.096.376.103.84833
401.704.084.029.021.754.146.284.676.546.389.983.645.676.482.641.457.120.920.480.407.10466
8026Info 247.310.769.186.488.306.440.493.305.677.95433
61.162.616.555.612.493.985.389.992.772.177.077.127.729.998.358.939.421.655.561.626.11665
8025Info 228.562.535.427.841.869.726.146.472.478.81533
52.240.832.601.203.469.830.544.252.552.896.269.825.525.244.503.896.430.210.623.804.22565
8022Info 121.835.770.286.544.029.000.701.461.776.97133
14.843.954.921.315.524.906.513.066.827.400.500.472.866.031.560.942.551.312.945.934.84165
6827Info 30.698.192.507.926.605.052.067.894.676.19332
942.379.023.253.721.145.582.536.575.330.464.033.575.635.285.541.127.352.320.973.24963
6249Info 3.144.120.818.001.204.461.096.631.877.16731
9.885.495.718.188.563.066.416.784.240.092.900.424.876.146.603.658.818.175.945.88961
6248Info 2.633.464.189.244.332.332.100.361.401.58631
6.935.133.636.032.308.614.662.012.098.578.758.902.102.664.168.032.306.363.315.39661
6247Info 2.522.532.318.987.074.601.528.732.897.80631
6.363.169.300.334.308.290.240.718.149.593.959.418.170.420.928.034.330.039.613.63661
6244Info 1.350.015.677.155.028.833.305.775.848.34131
1.822.542.328.564.351.039.723.668.198.972.798.918.663.279.301.534.658.232.452.28161
6243Info 1.193.375.854.875.511.918.605.120.376.07931
1.424.145.930.999.858.883.871.649.154.771.774.519.461.783.888.589.990.395.414.24161
6242Info 1.187.054.882.257.789.974.300.764.887.92931
1.409.099.293.492.055.619.115.210.663.950.593.660.125.119.165.502.943.929.909.04161
5268Info 725.657.034.735.673.961.274.833.643.72530
526.578.132.061.371.126.000.669.458.795.597.854.966.000.621.173.160.231.875.62560
5267Info 228.106.257.936.763.286.220.319.446.69530
52.032.464.909.913.183.706.998.947.070.107.074.989.960.738.131.990.946.423.02559
5265Info 130.303.161.860.097.498.924.329.036.46930
16.978.913.990.738.767.495.836.054.167.876.145.063.859.476.783.709.931.987.96159
5172Info Prime!    102.956.548.239.095.784.750.974.549.05130
10.600.050.825.309.257.335.789.552.697.979.625.598.753.375.290.352.805.000.60159
4793Info 69.697.645.997.692.395.446.140.967.17229
4.857.761.857.619.646.789.399.615.430.880.345.169.939.876.469.167.581.677.58458
4790Info 13.023.229.862.595.863.669.058.813.03129
169.604.516.054.008.678.101.490.292.111.292.094.101.876.800.450.615.406.96157
4789Info 11.219.988.402.958.206.841.478.502.21129
125.888.139.762.516.652.901.129.855.030.558.921.109.256.615.267.931.888.52157
3994Info 2.589.537.706.161.357.166.603.750.27428
6.705.705.531.631.423.370.133.506.726.276.053.310.733.241.361.355.075.07655
3992Info 1.373.512.530.649.258.635.292.477.60928
1.886.536.671.850.530.641.991.373.196.913.731.991.460.350.581.766.356.88155
3991Info 1.208.997.217.707.381.836.451.672.37128
1.461.674.272.424.190.432.753.231.165.611.323.572.340.914.242.724.761.64155
3978Info 1.103.866.360.472.916.946.437.760.98928
1.218.520.941.783.723.816.716.782.817.182.876.176.183.273.871.490.258.12155
3611Info 307.319.174.379.604.964.632.119.14327
94.445.074.941.362.044.498.184.066.966.048.189.444.026.314.947.054.44953
3610Info 259.534.999.916.657.357.224.172.37627
67.358.416.181.739.334.421.297.151.315.179.212.443.393.718.161.485.37653
3609Info 219.293.240.651.172.832.756.867.92227
48.089.525.395.293.201.014.489.455.855.498.441.010.239.259.352.598.08453
3606Info 122.063.831.551.139.898.460.740.72127
14.899.578.972.945.056.149.893.218.681.239.894.165.054.927.987.599.84153
3605Info 121.096.650.591.136.334.960.853.87927
14.664.398.784.391.760.063.516.738.283.761.536.006.719.348.789.346.64153
2975Info 64.897.400.105.515.621.177.314.68226
4.211.672.540.455.378.958.718.869.999.688.178.598.735.540.452.761.12452
2974Info 22.859.776.159.292.529.974.209.48526
522.569.366.052.959.132.741.539.313.935.147.231.959.250.663.965.22551
2673Info 4.275.548.277.509.699.161.443.65925
18.280.313.073.316.155.472.257.355.375.227.455.161.337.031.308.28150
2672Info 2.198.834.453.254.769.178.871.87225
4.834.872.952.820.199.705.196.890.986.915.079.910.282.592.784.38449
2671Info 2.107.561.207.005.423.534.632.46225
4.441.814.241.274.157.711.392.425.242.931.177.514.721.424.181.44449
2176Info 831.775.153.121.251.039.203.51424
691.849.905.349.880.612.384.525.525.483.216.088.943.509.948.19648
2175Info 722.956.456.358.957.313.434.53524
522.666.037.791.100.950.480.675.576.084.059.001.197.730.666.22548
2173Info 136.569.391.518.194.940.977.04124
18.651.198.699.650.016.285.556.865.558.261.005.699.689.115.68147
1940Info 25.886.574.803.023.261.617.72623
670.114.755.032.518.816.023.606.320.618.815.230.557.411.07645
1939Info 25.145.697.104.636.357.696.39423
632.306.082.878.117.302.583.050.385.203.711.878.280.603.23645
1938Info 20.706.193.716.777.033.766.26823
428.746.458.236.696.712.033.040.330.217.696.632.854.647.82445
1935Info 12.093.315.807.217.013.334.12123
146.248.287.213.084.882.816.757.618.288.480.312.782.842.64145
1556Info 3.823.177.109.095.314.778.62122
14.616.683.207.510.408.440.555.504.480.401.570.238.661.64144
1555Info 3.036.233.455.854.775.865.62322
9.218.713.598.451.835.185.192.915.815.381.548.953.178.12943
1554Info 2.211.007.906.320.264.673.87822
4.888.555.961.810.720.288.016.108.820.270.181.695.558.88443
1553Info 2.016.192.509.426.760.663.59822
4.065.032.235.068.578.387.346.437.838.758.605.322.305.60443
1551Info 1.121.621.184.033.823.231.53922
1.258.034.080.473.435.562.012.102.655.343.740.804.308.52143
1374Info 637.323.988.797.048.057.09821
406.181.866.696.179.837.389.983.738.971.696.668.181.60442
1373Info 263.397.138.410.176.890.08621
69.378.052.522.669.882.013.031.028.896.622.525.087.39641
1372Info 260.088.804.053.044.407.02621
67.646.185.993.742.928.767.376.782.924.739.958.164.67641
1371Info 229.359.782.235.085.482.22521
52.605.909.706.925.833.964.246.933.852.960.790.950.62541
1368Info 128.501.150.236.577.373.66921
16.512.545.612.123.429.216.861.292.432.121.654.521.56141
1082Info 25.686.162.978.506.292.36620
659.778.968.558.387.244.939.442.783.855.869.877.95639
950Info 2.282.211.769.458.230.80519
5.208.490.560.653.668.833.388.663.560.650.948.02537
947Info 1.363.859.210.193.543.29119
1.860.111.945.229.755.699.965.579.225.491.110.68137
741Info 795.559.265.009.384.10618
632.914.544.142.271.449.944.172.241.445.419.23636
740Info 633.856.150.760.638.65218
401.773.619.857.093.475.574.390.758.916.377.10436
739Info 404.099.764.753.665.98118
163.296.619.873.968.186.681.869.378.916.692.36136
738Info 306.950.094.269.977.05718
94.218.360.372.347.802.120.874.327.306.381.24935
737Info 228.138.929.476.341.40518
52.047.371.142.611.077.177.011.624.117.374.02535
736Info 207.254.460.945.174.61818
42.954.411.581.674.911.011.947.618.511.445.92435
734Info 135.772.344.267.730.05918
18.434.129.467.955.011.411.055.976.492.143.48135
640Info 40.447.213.778.058.76917
1.635.977.102.407.987.117.897.042.017.795.36134
637Info Prime Curios!    13.661.181.333.262.45917
186.627.875.420.278.656.872.024.578.726.68133
636Info 13.593.470.459.544.30917
184.782.439.134.503.767.305.431.934.287.48133
635Info 12.797.593.520.483.48117
163.778.399.915.520.777.025.519.993.877.36133
634Info 11.863.792.420.598.92917
140.749.570.599.060.595.060.995.075.947.04133
633Info 11.843.191.515.764.82117
140.261.185.279.083.838.380.972.581.162.04133
486Info 6.819.209.882.215.74216
46.501.623.417.708.833.880.771.432.610.56432
485Info 3.138.199.296.186.06716
9.848.294.822.582.726.272.852.284.928.48931
484Info 3.107.974.295.870.66316
9.659.504.223.792.743.472.973.224.059.56931
483Info 3.066.446.727.654.24316
9.403.095.533.541.415.141.453.355.903.04931
482Info 2.564.053.868.197.73416
6.574.372.239.019.762.679.109.322.734.75631
481Info 2.201.019.508.986.47816
4.844.486.878.939.076.709.398.786.844.48431
415Info 314.155.324.482.86715
98.693.567.900.935.453.900.976.539.68929
414Info 210.786.628.549.53815
44.431.002.775.280.908.257.720.013.44429
411Info 129.610.990.752.56915
16.799.008.923.862.526.832.980.099.76129
310Info 98.275.825.201.58714
9.658.137.819.052.882.509.187.318.56928
309Info 69.800.670.077.02814
4.872.133.543.202.112.023.453.312.78428
308Info 30.395.080.190.57314
923.860.899.791.363.197.998.068.32927
264Info 6.360.832.925.89813
40.460.195.511.188.111.559.106.40426
263Info 3.069.306.930.69313
9.420.645.034.800.084.305.460.24925
262Info 2.634.812.417.86413
6.942.236.477.330.337.746.322.49625
261Info 2.149.099.165.35813
4.618.627.222.542.452.227.268.16425
258Info 1.349.465.117.84113
1.821.056.104.269.624.016.501.28125
192Info 128.817.084.66912
16.593.841.302.620.314.839.56123
191Info 112.247.658.96112
12.599.536.942.224.963.599.52123
190Info 111.283.619.36112
12.384.043.938.083.934.048.32123
178Info 101.116.809.85112
10.224.609.234.443.290.642.20123
162Info 83.163.115.48611
6.916.103.777.337.773.016.19622
161Info 30.693.069.30711
942.064.503.484.305.460.24921
160Info 30.101.273.64711
906.086.675.171.576.680.60921
159Info 22.865.150.13511
522.815.090.696.090.518.22521
156Info 13.579.355.05911
184.398.883.818.388.893.48121
137Info 10.207.355.54911
104.190.107.303.701.091.40121
130Info 10.106.064.39911
102.132.537.636.735.231.20121
113Info 2.481.623.25410
6.158.453.974.793.548.51619
112Info 2.062.386.21810
4.253.436.912.196.343.52419
95Info 306.930.6939
94.206.450.305.460.24917
70Info 64.030.6488
4.099.923.883.299.90416
69Info Prime!    30.001.2538
900.075.181.570.00915
67Info 12.866.6698
165.551.171.155.56115
66Info 12.028.2298
144.678.292.876.44115
65Info 11.129.3618
123.862.676.268.32115
56Info 3.069.3077
9.420.645.460.24913
55Info 2.294.6757
5.265.533.355.62513
54Info 2.012.7487
4.051.154.511.50413
51Info 1.270.8697
1.615.108.015.16113
44Info 1.042.1517
1.086.078.706.80113
37Info 798.6446
637.832.238.73612
31Info 30.6935
942.060.2499
30Info 24.8465
617.323.7169
29Info 22.8655
522.808.2259
20Info 2.6364
6.948.4967
19Info 2.2854
5.221.2257
15Info 8363
698.8966
14Info Prime!    3073
94.2495
13Info 2643
69.6965
7Info 262
6763


Contributions

[ October 1996 ]
During this period PDG was making a table of palindromic squares from scratch himself, hoping to find larger numbers.
He succeeded beginning from [308] !
(Un)fortunately soon afterwards this record was smashed by submitters like Graham Lyons, Pete Leadbetter
and Mike Bennett who sent in numbers of even greater calibre ! Observe that the top three palindromic square
submitters were all British ! Was I detecting a new pattern here ?

[ April 29, 1997 ]
Many thanks go to Graham Lyons (email) who submitted the following number of 36 digits .
See index number [741].
The first even palindromic square of length 32 was also submitted by Graham Lyons
See index number [486].

[ June 27, 1997 ]
Pete Leadbetter submitted also this number of 44 digits.
See index number [1556].
Pete Leadbetter informs me that he checked for 40 digit palindromes and found none.
He's convinced that there aren't any 38's since his 'old' program searched that far.
There are no more 44 digit squares that are palindromic other than the above one.
In his letter he says also that for the case of 42 digits it will take nearly as long (12 days on his PC !)
and leave the search for someone else with enough free computer timecycles.

Congratulations also to Mike Bennett (email) who submitted the following even digit palindromic squares of 36, 42 and 48 digits
as recently as [ November 29, 1997 ].
See index number [740].
and the following a few days later [ December 5, 1997 ].
See index number [1374].
and finally this actual record number sent [ January 13, 1998 ].
See index number [2176].
Mike found these palindromes using an algorithm in C++ that runs on an Acorn RiscPC with a StrongARM processor.
He also confirms that there are no 38 or 40 digit solutions.
Regarding the time to test for all 46 and 48 digit palindromes Mike wrote me that he estimates this needs about 150 days.
He tests for numbers of the form 4n and 4n-2 in parallel. The 48 digit record palindrome was a fluke that turned up in less than 1 day !

[ February 2001 ]
William Rex Marshall calculated and posted all the Palindromic Squares
of the sporadic type smaller than 1040. Three more palindromic squares sofar unknown to me
could be extracted and added to the list.
Referenced index numbers [737], [739] and [950]
The second one is of even length 36, which surprised me as I thought that all these
would have been detected already... Somehow a few could escape detection.

[ May 20, 2001 ]
Many thanks go to Pete Leadbetter who submitted the largest palindromic square with a non_palindromic non_core base of 52 digits.
See index number [2975].
“ Pete wrote that this square was found using an Athlon 1200 MHZ PC programmed in Powerbasic about a third
of the way into a 23 day search. The algorithm is basically the same as the one used to find the previous 44 digit number.
A faster machine plus the more efficient compilation that Powerbasic affords allowed the 52 digit range to be looked at within a
'not so mind boggling' time. By the way, the find reported to you was the result of a 'burn in' for my new PC, as was my previous
44 digit result! Also I'd recommend Powerbasic for number crunching, it's seriously quick!
Pete remarks that of the set of EVEN length palindromic squares none of them seems to end in digit 5. Is there a proof for this?
If the base ends in 5 then the square must end in 25. Perhaps this reduces the freedom of one of the digits thereby reducing
the likelyhood of such a number. Or are there other number theory reasons? ”

[ March 30-31, 2002 ]
Feng Yuan (email) from Washington State, USA, sent five new sporadic palindromic squares of 41 and 43 digits.
Website address Palindromic Square Numbers
Referenced index numbers [1368], [1371], [1372], [1373], [1551], [1553], [1554] and [1555].
“ The palindromic square number searching algorithm starts by assuming a generic quadratic formula F(n) = An^2 + Bn + C, where A, B, and C are
non-negative integer numbers. Now the code tries to find palindrome number when n is within a certain range 'from' to 'to'. It divides the range into
subranges such that for the first and last n within the range F(n) have the same length and same first digit. For that range, it searches all the last
number of n within 0..9 such that the latest digit of F(n) matches the first digit of F(n). Now we know when n is within that range, and when n is of the
form 10 * m + f, the first and last digit of F(n) are always the same (f). After that the code substitute n = 10 * m + f into F(n), remove the highest
digit, divide by 10 to remove the last digit to get F(m). The routine is called recursively to work on the new formula until the range is small
enough to work with using an iterative algorithm.
I'm using a 1.4Ghz Pentium 4 machine, running Windows XP. Program is written in Microsoft Visual C++ 6.0. The time the algorithm needs to
complete search for n-digit palindrome squares is around O(10^(n/4)), that is to say time increases 10 times for every 4 digits. Using the current
code, I estimate that it will take around 200 days to complete search all the 52-digit numbers. I hope I can reduce that to something like 50 days. ”

Feng Yuan (email) from Washington State, USA, sent four new sporadic palindromic squares of 45 digits [ April 19, 2002 ].
Website address Palindromic Square Numbers
Referenced index numbers [1935], [1938], [1939] and [1940].
“ The complete search for 45-digit palindromic numbers can be completed
in around 30 hours. How is that done? I doubled the speed of my code and
got a 2.0 Ghz dual-CPU machine. ”

[ May 14, 2002 ]
Hidenori Takahashi (email) from Japan, sent in a new sporadic palindromic square of 48 digits.
See index number [2175].
" My name is Takahashi, and I live in Osaka. I am untrained and doesn't know any English at all, so ask a Japanese to translate this message. ( ps. done so! )
Since I love mathematics, I was very interested when I saw these numbers. In Japanese we call these palindromic squares 'kaibunheihousuu',
and my program is an improvement of Assembler code made 15 years ago. Its efficiency is still bad, but because I investigate very thoroughly,
there are no unexpected slovenly errors. Palindromic squares will arise 66 days after the program starts when it is ready of course. I expect to
finish it the second half of July."

[ January 1-2, 2008 ]
Feng Yuan (email) from Washington State, USA, sent record palindromic squares with odd numbers 47 & 49 of digits.
Website address Palindromic Square Numbers
See index number [2173], [2671] and [2672].
“ I found a new one last night using my 8-core machine.
Happy New Year. "
" The machine I'm using is Dell Precision PWS690, 2.33GHz. It's dual QuadCore, so 8 cores in total.
I'm trying to use all 8 cores at the same time to boost performance. The complete search for 47 digits
took 35,540 seconds (clock time), around 10 hours.
It's still 32-bit programs. 64-bit would be definitely faster, but I do not have it installed on my machine.
Searching 48 digits came back empty. Now I'm in the middle of searching for 49 digits. ”

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, sent a palindromic square with even numbers 50 of digits.

[ January 16, 2008 ]
Feng Yuan (email) from Washington State, USA, sent a record sporadic palindromic square 53 of digits.

[ November 11, 2022 ]
Robert Xiao (email) added a new square sporadic palindromic.

“ I've been working on a new program. It harnesses the GPU to crunch palindromes over 10x faster than my previous program,
and that on my six-year-old spare-parts desktop. It'll probably be significantly faster with new hardware.
I'm one hour into my run of 51-digit square palindromes, and I have a new SSP:
See index nr [2974].
It's no world record, but we're tantalizingly close. 50 took just 72 minutes with this program,
so 56 should take a day or two for me (and much less with faster hardware - which David Griffeath has!)
Best,
Robert ”

[ November 13, 2022 ]
Robert Xiao (email) added all the squares from [2674] up to [3611]
whereby [2974] is a new square sporadic palindrome just like [3610] and [3611].

“ Unfortunately, there are no SP's of length 54.

More excitingly, I ran my program at length 55. Unfortunately, a few hours into my run,
my computer unexpectedly rebooted - seems to have been one of those cursed Windows updates.
Nevertheless, after just three hours, my program had spat out a brand new, world record (!)
SSP of length 55.
See provisional index number [3994].

Best,
Robert ”

[ November 19, 2022 ]
Robert Xiao (email)

“ Happy to report that 58 is not void :)
See provisional index number [4793]

Robert ”

[ November 24, 2022 ]
David Griffeath (email)
is the first person to cross the 60+ digit barrier by sending in this record palindromic square [6244]
As always he did this with the help of Robert Xiao's very efficient CUDA gpu program.

[ December 4, 2022 ]
David Griffeath (email)
finished the exhaustive 61-digit search. The results are bountiful since six new palindromic squares were found.
That was ago from length 21 !
Consult the official index numbers [6242], [6243], [6244], [6247], [6248] and [6249].
Using Robert Xiao's program 'CUDApalin' it took about 10 days, 4 hours and 21 minutes to handle length 61.

[ December 10, 2022 ]
David Griffeath (email) wrote
“ Over the past week, Robert and I combined forces to complete an exhaustive search for 62-digit SSPs.
The result, in current American vernacular: a complete and total nothingburger.
This time-consuming result of minimal substance has two silver linings:
1) The current record SSP remains intact;
2) You can extend your complete listings and tables for both SPs and SSPs through length 62 with minimum effort.”

[ January 29, 2023 ]
David Griffeath (email) wrote
“ The exhaustive search of 63 digits is complete and there is a unique, new SSP, discovered on January 2 [6827].
I'm telling myself it's cool there's only one of 1063–1062 integers with the magic property.”

[ February 13, 2023 ]
Robert Xiao (email) informs
“ The search for palindromic squares (SPs) at length 64 is concluded after a grueling 27.9 days of solid GPU time.
No SPs were found, so the 63 record still stands. ”

[ May 12, 2023 ]
David Griffeath (email) wrote
“ I made one last push for the 65-digit case, Tyche (the goddess of chance) acceded to my
stubbornness and let me discover this new SSP on May 12 [8022] shortly after midnight.”

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