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EMBED
Subsets of Palindromic Squares
rood All Squares up to length 31 rood The Sporadic Squares rood
rood Palindromic Squares rood Factorization


Itinerary
Sporadic Palindromic Squares of EVEN length
Score 'subset E' = 22   
Sporadic Palindromic Squares of ODD length
Score 'subset O' = 95   
Ultra Square Palindromes (by D. Griffeath)
Score 'subset U' = 21   
Pari/gp script for generating all the NSSP's
→ Non Sporadic Square Palindromes   


 PLAIN TEXT SQUARES      PLAIN TEXT SSP 

Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The subsets of palindromic squares are categorised as follows :
separator
%N n^2 is a palindromic square of sporadic type. under A059744.
%N Palindromic squares of sporadic type. under A059745.
separator
%N Square is a palindrome with even number of digits. under A016113.
%N Palindromic squares with an even number of digits. under A027829.
separator
%N Square is a palindrome with odd number of digits. under A028816.
%N Palindromic squares with odd number of digits. under A028817.
separator
%N Nonpalindromic & "non-core" numbers that when squared give palindromes
with odd number of digits. under A016106.
%N Palindromic squares with odd number of digits and nonpalindromic & "non-core"
square roots. under A028818.
%N Asymmetric families of palindromic squares. under A007573.
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.


Sporadic Palindromic Squares of EVEN length
Sporadic Palindromic Squares of ODD length
Score 'subset O' = 95   
Ultra Square Palindromes (by D. Griffeath)
Score 'subset U' = 21   
Pari/gp script for generating all the NSSP's
→ Non Sporadic Square Palindromes   


Index
Number		Offic.
nbr		 BasenumberLength
Palindromic Square Number of EVEN lengthLength
   
See also Sloane's A059744 and A059745
228028 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
218027 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
205268 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
194793 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
182975 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
172673 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
162176 831.775.153.121.251.039.203.51424
691.849.905.349.880.612.384.525.525.483.216.088.943.509.948.19648
152175 722.956.456.358.957.313.434.53524
522.666.037.791.100.950.480.675.576.084.059.001.197.730.666.22548
141556 3.823.177.109.095.314.778.62122
14.616.683.207.510.408.440.555.504.480.401.570.238.661.64144
131374 637.323.988.797.048.057.09821
406.181.866.696.179.837.389.983.738.971.696.668.181.60442
12741 795.559.265.009.384.10618
632.914.544.142.271.449.944.172.241.445.419.23636
11740 633.856.150.760.638.65218
401.773.619.857.093.475.574.390.758.916.377.10436
10739 404.099.764.753.665.98118
163.296.619.873.968.186.681.869.378.916.692.36136
9640 40.447.213.778.058.76917
1.635.977.102.407.987.117.897.042.017.795.36134
8486 6.819.209.882.215.74216
46.501.623.417.708.833.880.771.432.610.56432
7310 98.275.825.201.58714
9.658.137.819.052.882.509.187.318.56928
6309 69.800.670.077.02814
4.872.133.543.202.112.023.453.312.78428
5264 6.360.832.925.89813
40.460.195.511.188.111.559.106.40426
4162 83.163.115.48611
6.916.103.777.337.773.016.19622
370 64.030.6488
4.099.923.883.299.90416
237 798.6446
637.832.238.73612
115 8363
698.8966


Sporadic Palindromic Squares of ODD length
Sporadic Palindromic Squares of EVEN length
Score 'subset E' = 22   
Ultra Square Palindromes (by D. Griffeath)
Score 'subset U' = 21   
Pari/gp script for generating all the NSSP's
→ Non Sporadic Square Palindromes   


Index
Number		Offic.
nbr		 BasenumberLength
Palindromic Square Numbers of ODD lengthLength
   
See also Sloane's A059744 and A059745
958026 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
948025 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
938022 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
926827 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
916249 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
906248 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
896247 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
886244 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
876243 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
866242 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
855267 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
845265 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
835172 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
824790 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
814789 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
803994 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
793992 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
783991 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
773978 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
763611 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
753610 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
743609 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
733606 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
723605 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
712974 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
702672 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
692671 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
682173 136.569.391.518.194.940.977.04124
18.651.198.699.650.016.285.556.865.558.261.005.699.689.115.68147
671940 25.886.574.803.023.261.617.72623
670.114.755.032.518.816.023.606.320.618.815.230.557.411.07645
661939 25.145.697.104.636.357.696.39423
632.306.082.878.117.302.583.050.385.203.711.878.280.603.23645
651938 20.706.193.716.777.033.766.26823
428.746.458.236.696.712.033.040.330.217.696.632.854.647.82445
641935 12.093.315.807.217.013.334.12123
146.248.287.213.084.882.816.757.618.288.480.312.782.842.64145
631555 3.036.233.455.854.775.865.62322
9.218.713.598.451.835.185.192.915.815.381.548.953.178.12943
621554 2.211.007.906.320.264.673.87822
4.888.555.961.810.720.288.016.108.820.270.181.695.558.88443
611553 2.016.192.509.426.760.663.59822
4.065.032.235.068.578.387.346.437.838.758.605.322.305.60443
601551 1.121.621.184.033.823.231.53922
1.258.034.080.473.435.562.012.102.655.343.740.804.308.52143
591373 263.397.138.410.176.890.08621
69.378.052.522.669.882.013.031.028.896.622.525.087.39641
581372 260.088.804.053.044.407.02621
67.646.185.993.742.928.767.376.782.924.739.958.164.67641
571371 229.359.782.235.085.482.22521
52.605.909.706.925.833.964.246.933.852.960.790.950.62541
561368 128.501.150.236.577.373.66921
16.512.545.612.123.429.216.861.292.432.121.654.521.56141
551082 25.686.162.978.506.292.36620
659.778.968.558.387.244.939.442.783.855.869.877.95639
54950 2.282.211.769.458.230.80519
5.208.490.560.653.668.833.388.663.560.650.948.02537
53947 1.363.859.210.193.543.29119
1.860.111.945.229.755.699.965.579.225.491.110.68137
52738 306.950.094.269.977.05718
94.218.360.372.347.802.120.874.327.306.381.24935
51737 228.138.929.476.341.40518
52.047.371.142.611.077.177.011.624.117.374.02535
50736 207.254.460.945.174.61818
42.954.411.581.674.911.011.947.618.511.445.92435
49734 135.772.344.267.730.05918
18.434.129.467.955.011.411.055.976.492.143.48135
48637 13.661.181.333.262.45917
186.627.875.420.278.656.872.024.578.726.68133
47636 13.593.470.459.544.30917
184.782.439.134.503.767.305.431.934.287.48133
46635 12.797.593.520.483.48117
163.778.399.915.520.777.025.519.993.877.36133
45634 11.863.792.420.598.92917
140.749.570.599.060.595.060.995.075.947.04133
44633 11.843.191.515.764.82117
140.261.185.279.083.838.380.972.581.162.04133
43485 3.138.199.296.186.06716
9.848.294.822.582.726.272.852.284.928.48931
42484 3.107.974.295.870.66316
9.659.504.223.792.743.472.973.224.059.56931
41483 3.066.446.727.654.24316
9.403.095.533.541.415.141.453.355.903.04931
40482 2.564.053.868.197.73416
6.574.372.239.019.762.679.109.322.734.75631
39481 2.201.019.508.986.47816
4.844.486.878.939.076.709.398.786.844.48431
38415 314.155.324.482.86715
98.693.567.900.935.453.900.976.539.68929
37414 210.786.628.549.53815
44.431.002.775.280.908.257.720.013.44429
36411 129.610.990.752.56915
16.799.008.923.862.526.832.980.099.76129
35308 30.395.080.190.57314
923.860.899.791.363.197.998.068.32927
34263 3.069.306.930.69313
9.420.645.034.800.084.305.460.24925
33262 2.634.812.417.86413
6.942.236.477.330.337.746.322.49625
32261 2.149.099.165.35813
4.618.627.222.542.452.227.268.16425
31258 1.349.465.117.84113
1.821.056.104.269.624.016.501.28125
30192 128.817.084.66912
16.593.841.302.620.314.839.56123
29191 112.247.658.96112
12.599.536.942.224.963.599.52123
28190 111.283.619.36112
12.384.043.938.083.934.048.32123
27178 101.116.809.85112
10.224.609.234.443.290.642.20123
26161 30.693.069.30711
942.064.503.484.305.460.24921
25160 30.101.273.64711
906.086.675.171.576.680.60921
24159 22.865.150.13511
522.815.090.696.090.518.22521
23156 13.579.355.05911
184.398.883.818.388.893.48121
22137 10.207.355.54911
104.190.107.303.701.091.40121
21130 10.106.064.39911
102.132.537.636.735.231.20121
20113 2.481.623.25410
6.158.453.974.793.548.51619
19112 2.062.386.21810
4.253.436.912.196.343.52419
1895 306.930.6939
94.206.450.305.460.24917
1769 30.001.2538
900.075.181.570.00915
1667 12.866.6698
165.551.171.155.56115
1566 12.028.2298
144.678.292.876.44115
1465 11.129.3618
123.862.676.268.32115
1356 3.069.3077
9.420.645.460.24913
1255 2.294.6757
5.265.533.355.62513
1154 2.012.7487
4.051.154.511.50413
1051 1.270.8697
1.615.108.015.16113
944 1.042.1517
1.086.078.706.80113
831 30.6935
942.060.2499
730 24.8465
617.323.7169
629 22.8655
522.808.2259
520 2.6364
6.948.4967
419 2.2854
5.221.2257
314 3073
94.2495
213 2643
69.6965
17 262
6763


Ultra Square Palindromes
Sporadic Palindromic Squares of EVEN length
Score 'subset E' = 22   
Sporadic Palindromic Squares of ODD length
Score 'subset O' = 95   
Pari/gp script for generating all the NSSP's
→ Non Sporadic Square Palindromes   
David Griffeath an emeritus professor from the University of Wisconsin - Madison
having fun with square palindromes in his retirement. He has found a cute special case and wonder whether
you know of any previous discussion of numbers with these three additional properties:

(i) all digits are positive;
(ii) the sum of digits is a perfect square;
(iii) the product of digits is a perfect square.

He has found 21 examples:
16 with binary (all digits 0 or 1) square roots which can be identified without computer search,
1 from the Even Root family (the very smallest USP),
1 with an Asymmetric basenumber containing the digits 0, 1 or 9, and
3 sporadic entries from the search tables.

David also conjectures that these are all the examples with extra properties (i)-(iii).
In addition, he conjectures that there are only finitely many square palindromes with all digits positive.


Index
Number		Offic.
nbr		 BasenumberLength
Ultra Square PalindromesLength
   
By David Griffeath
who conjectures that there are only 21 USP's (see PDF article)
21741 795.559.265.009.384.10618
632.914.544.142.271.449.944.172.241.445.419.23636
20628 11.100.100.100.100.11117
123.212.222.232.242.494.242.232.222.212.32133
19410 111.100.010.001.11115
12.343.212.222.246.964.222.221.234.32129
18408 111.010.010.010.11115
12.323.222.322.444.944.422.322.232.32129
17406 111.001.010.100.11115
12.321.224.243.244.944.234.242.212.32129
16397 110.101.010.101.01115
12.122.232.425.262.926.252.423.222.12129
15257 1.111.001.001.11113
1.234.323.224.469.644.223.234.32125
14255 1.110.101.010.11113
1.232.324.252.649.462.524.232.32125
13253 1.110.011.100.11113
1.232.124.642.369.632.464.212.32125
12248 1.101.101.011.01113
1.212.423.436.449.446.343.242.12125
11246 1.101.011.101.01113
1.212.225.444.549.454.445.222.12125
10156 13.579.355.05911
184.398.883.818.388.893.48121
9155 11.110.101.11111
123.434.346.696.643.434.32121
8153 11.101.110.11111
123.234.645.696.546.432.32121
7148 11.011.111.01111
121.244.565.696.565.442.12121
6113 2.481.623.25410
6.158.453.974.793.548.51619
592 111.111.1119
12.345.678.987.654.32117
490 111.091.1119
12.341.234.943.214.32117
324 11.0115
1.212.42.12110
217 1.1114
1.234.3217
112 2123
44.9445


Non Sporadic Square Palindromes → a Pari/gp script generator
Sporadic Palindromic Squares of EVEN length
Score 'subset E' = 22   
Sporadic Palindromic Squares of ODD length
Score 'subset O' = 95   
Ultra Square Palindromes (by D. Griffeath)
Score 'subset U' = 21   


I am much indebted to David Griffeath, my palindromic squares coach, who encouraged me to program
for these nssp's. I quote “Since the designs for each of the four infinite families are quite simple, it would
be better to use a program to enumerate those to any desired size far exceeding those of any complete list
of sporadics that will ever be feasible. Even the asymmetric family should be quite doable with case
checking for the palindrome property.”
So, allow me to present to you my final NSSP (Non Sporadic Square Palindromes) full list generator written
in Pari/gp code. This program takes care of all the categories that are non sporadic. Copy/paste the code
hereunder in your favorite texteditor and save as 'nssp.gp'. It takes away my burden to list them in ever longer
boring tables. The four categories encompasses

1. the EVEN root family
2. the BINARY root family
3. the TERNARY root family
4. the ASYMMETRIC family

And you'll find all those entries per basenumber length in the file 'nssp.txt' in an very orderly fashion.
You can read this file with any decent texteditor. On screen you will see alas an unordered list scrolling by.
Some notes
Change with a texteditor the value of the variable bnl (basenumber length (⩾2), set at 31) to whatever you need.
Make sure the path to the folder 'C:/pari/' exists otherwise an error occurs. Change its name at will of course.

Pscr() = of++; print(of," ",b," {BL=",bnl,"} ",bq," {PL=",#digits(bq),"}");
{

bnl=31;

of=0; pr=(bnl/2!=bnl\2); if(pr, b=2*10^(bnl-1)+10^(bnl\2)+2; silo=List(b);
bq=b^2; Pscr(), silo=List() );

bg=3^(bnl\2-1+pr); for(i=bg, bg*2,
li=digits(i,3); b=fromdigits(concat(li,Vecrev(li[1..#(li)-pr]))); bq=b^2;
if(digits(bq)==Vecrev(digits(bq)), Pscr(); listput(silo,b) ));

bg=2^(bnl\2-1+pr); for(i=bg, bg*2,
li=binary(i); b=fromdigits(concat(li,Vecrev(li[1..#(li)-pr])));
for(y=0,bnl\4, b=digits(b); b[bnl\2+1-y]=9; b[bnl\2-y]=0; b=fromdigits(b); bq=b^2;
if(digits(bq)==Vecrev(digits(bq)), i++; Pscr(); listput(silo,b) )));

listsort(silo); of=0; ps="c:/pari/nssp.txt"; write1(ps, "\n");
for(s=1,#(silo), b=silo[s]; bq=b^2; of++;
write(ps, of," (PG4) ",b," {BL=",bnl,"} [",bq,"] {PL=",#digits(bq),"}") );
}

Following is a example output from 'nssp.txt' with bnl = 9.
Note the two Asymmetric solutions highlighted in yellow, the four with a Ternary Root highlighted in blue and
the last two from the Even Root family in green ! The rest in white are from the Binary Root family.

1 (PG4) 100000001 {BL=9} [10000000200000001] {PL=17}
2 (PG4) 100010001 {BL=9} [10002000300020001] {PL=17}
3 (PG4) 100020001 {BL=9} [10004000600040001] {PL=17}
4 (PG4) 100101001 {BL=9} [10020210401202001] {PL=17}
5 (PG4) 100111001 {BL=9} [10022212521222001] {PL=17}
6 (PG4) 100121001 {BL=9} [10024214841242001] {PL=17}
7 (PG4) 101000101 {BL=9} [10201020402010201] {PL=17}
8 (PG4) 101010101 {BL=9} [10203040504030201] {PL=17}
9 (PG4) 101020101 {BL=9} [10205060806050201] {PL=17}
10 (PG4) 101101101 {BL=9} [10221432623412201] {PL=17}
11 (PG4) 101111101 {BL=9} [10223454745432201] {PL=17}
12 (PG4) 110000011 {BL=9} [12100002420000121] {PL=17}
13 (PG4) 110010011 {BL=9} [12102202520220121] {PL=17}
14 (PG4) 110020011 {BL=9} [12104402820440121] {PL=17}
15 (PG4) 110091011 {BL=9} [12120030703002121] {PL=17}
16 (PG4) 110101011 {BL=9} [12122232623222121] {PL=17}
17 (PG4) 110111011 {BL=9} [12124434743442121] {PL=17}
18 (PG4) 111000111 {BL=9} [12321024642012321] {PL=17}
19 (PG4) 111010111 {BL=9} [12323244744232321] {PL=17}
20 (PG4) 111091111 {BL=9} [12341234943214321] {PL=17}
21 (PG4) 111101111 {BL=9} [12343456865434321] {PL=17}
22 (PG4) 111111111 {BL=9} [12345678987654321] {PL=17}
23 (PG4) 200000002 {BL=9} [40000000800000004] {PL=17}
24 (PG4) 200010002 {BL=9} [40004000900040004] {PL=17}

The following shorter Pari programs produces separate screenoutput for each of the four families B, T, A, E.
Replace the blue value of bnl with the digitlength of the basenumber of your choice.

Fist comes the Binary Family [ B ]

{
bnl=33;
of=0; pr=(bnl/2!=bnl\2); bg=2^(bnl\2-1+pr);
for(i=bg, bg*2-1, 
	li=binary(i); b=fromdigits(concat(li,Vecrev(li[1..#(li)-pr]))); bq=b^2;
	if(digits(bq)==Vecrev(digits(bq)), of++; print(of," ",b," {BL=",bnl,"} ",bq," {PL=",#digits(bq),"}")) );
}

Next comes the Ternary Family [ T ]

Pscr() = of++; b=fromdigits(b); print(of," ",b," {BL=",bnl,"} ",b^2," {PL=",#digits(b^2),"}");
{
bnl=33; if((-1)^bnl==-1, b=digits(10^(bnl-1)+2*10^(bnl\2)+1); of=0; Pscr(); a=1;
for(i=bnl\2+2, bnl-1, b=digits(b); b[i]=1; b[i-2*a]=1;
Pscr(); b=digits(b); b[i]=0; b[i-2*a]=0; a+=1; b=fromdigits(b) ),
print("No solution if bnl (=",bnl,") is even") );
}

After this allow me to present the Asymmetric Family [ A ]

{
bnl=33; of=0; pr=(bnl/2!=bnl\2); bg=2^(bnl\2-1+pr);
for(i=bg, bg*2-1, 
li=binary(i); b=fromdigits(concat(li,Vecrev(li[1..#(li)-pr]))); 
for(y=0,bnl\4, b=digits(b); b[bnl\2+1-y]=9; b[bnl\2-y]=0; b=fromdigits(b); bq=b^2;
if(digits(bq)==Vecrev(digits(bq)), of++; i++; print(of," ",b," {BL=",bnl,"} ",bq," {PL=",#digits(bq),"}")) ));
}

And finally the Even Root Family [ E ]

Pscr() = of++; b=fromdigits(b); print(of," ",b," {BL=",bnl,"} ",b^2," {PL=",#digits(b^2),"}");
{
bnl=33; b=digits(2*10^(bnl-1)+2); of=0; Pscr();
if((bnl/2!=bnl\2), b=digits(b); b[bnl\2+1]=1; Pscr() );
}

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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com