Itinerary
| → Non Sporadic Square Palindromes |
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 :
%N n^2 is a palindromic square of sporadic type. under A059744.
%N Palindromic squares of sporadic type. under A059745.
%N Square is a palindrome with even number of digits. under A016113.
%N Palindromic squares with an even number of digits. under A027829.
%N Square is a palindrome with odd number of digits. under A028816.
%N Palindromic squares with odd number of digits. under A028817.
%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.

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
| → Non Sporadic Square Palindromes |
Index Number | Offic. nbr |
Basenumber | Length |
Palindromic Square Number of EVEN length | Length |
| | |
See also Sloane's A059744 and A059745 |
22 | 8028 |
982.503.990.036.767.976.718.486.749.830.913 | 33 |
965.314.090.438.169.467.661.632.561.053.978.879.350.165.236.166.764.961.834.090.413.569 | 66 |
21 | 8027 |
633.801.296.960.665.546.843.096.376.103.848 | 33 |
401.704.084.029.021.754.146.284.676.546.389.983.645.676.482.641.457.120.920.480.407.104 | 66 |
20 | 5268 |
725.657.034.735.673.961.274.833.643.725 | 30 |
526.578.132.061.371.126.000.669.458.795.597.854.966.000.621.173.160.231.875.625 | 60 |
19 | 4793 |
69.697.645.997.692.395.446.140.967.172 | 29 |
4.857.761.857.619.646.789.399.615.430.880.345.169.939.876.469.167.581.677.584 | 58 |
18 | 2975 |
64.897.400.105.515.621.177.314.682 | 26 |
4.211.672.540.455.378.958.718.869.999.688.178.598.735.540.452.761.124 | 52 |
17 | 2673 |
4.275.548.277.509.699.161.443.659 | 25 |
18.280.313.073.316.155.472.257.355.375.227.455.161.337.031.308.281 | 50 |
16 | 2176 |
831.775.153.121.251.039.203.514 | 24 |
691.849.905.349.880.612.384.525.525.483.216.088.943.509.948.196 | 48 |
15 | 2175 |
722.956.456.358.957.313.434.535 | 24 |
522.666.037.791.100.950.480.675.576.084.059.001.197.730.666.225 | 48 |
14 | 1556 |
3.823.177.109.095.314.778.621 | 22 |
14.616.683.207.510.408.440.555.504.480.401.570.238.661.641 | 44 |
13 | 1374 |
637.323.988.797.048.057.098 | 21 |
406.181.866.696.179.837.389.983.738.971.696.668.181.604 | 42 |
12 | 741 |
795.559.265.009.384.106 | 18 |
632.914.544.142.271.449.944.172.241.445.419.236 | 36 |
11 | 740 |
633.856.150.760.638.652 | 18 |
401.773.619.857.093.475.574.390.758.916.377.104 | 36 |
10 | 739 |
404.099.764.753.665.981 | 18 |
163.296.619.873.968.186.681.869.378.916.692.361 | 36 |
9 | 640 |
40.447.213.778.058.769 | 17 |
1.635.977.102.407.987.117.897.042.017.795.361 | 34 |
8 | 486 |
6.819.209.882.215.742 | 16 |
46.501.623.417.708.833.880.771.432.610.564 | 32 |
7 | 310 |
98.275.825.201.587 | 14 |
9.658.137.819.052.882.509.187.318.569 | 28 |
6 | 309 |
69.800.670.077.028 | 14 |
4.872.133.543.202.112.023.453.312.784 | 28 |
5 | 264 |
6.360.832.925.898 | 13 |
40.460.195.511.188.111.559.106.404 | 26 |
4 | 162 |
83.163.115.486 | 11 |
6.916.103.777.337.773.016.196 | 22 |
3 | 70 |
64.030.648 | 8 |
4.099.923.883.299.904 | 16 |
2 | 37 |
798.644 | 6 |
637.832.238.736 | 12 |
1 | 15 |
836 | 3 |
698.896 | 6 |
Sporadic Palindromic Squares of ODD length
| → Non Sporadic Square Palindromes |
Index Number | Offic. nbr |
Basenumber | Length |
Palindromic Square Numbers of ODD length | Length |
| | |
See also Sloane's A059744 and A059745 |
95 | 8026 |
247.310.769.186.488.306.440.493.305.677.954 | 33 |
61.162.616.555.612.493.985.389.992.772.177.077.127.729.998.358.939.421.655.561.626.116 | 65 |
94 | 8025 |
228.562.535.427.841.869.726.146.472.478.815 | 33 |
52.240.832.601.203.469.830.544.252.552.896.269.825.525.244.503.896.430.210.623.804.225 | 65 |
93 | 8022 |
121.835.770.286.544.029.000.701.461.776.971 | 33 |
14.843.954.921.315.524.906.513.066.827.400.500.472.866.031.560.942.551.312.945.934.841 | 65 |
92 | 6827 |
30.698.192.507.926.605.052.067.894.676.193 | 32 |
942.379.023.253.721.145.582.536.575.330.464.033.575.635.285.541.127.352.320.973.249 | 63 |
91 | 6249 |
3.144.120.818.001.204.461.096.631.877.167 | 31 |
9.885.495.718.188.563.066.416.784.240.092.900.424.876.146.603.658.818.175.945.889 | 61 |
90 | 6248 |
2.633.464.189.244.332.332.100.361.401.586 | 31 |
6.935.133.636.032.308.614.662.012.098.578.758.902.102.664.168.032.306.363.315.396 | 61 |
89 | 6247 |
2.522.532.318.987.074.601.528.732.897.806 | 31 |
6.363.169.300.334.308.290.240.718.149.593.959.418.170.420.928.034.330.039.613.636 | 61 |
88 | 6244 |
1.350.015.677.155.028.833.305.775.848.341 | 31 |
1.822.542.328.564.351.039.723.668.198.972.798.918.663.279.301.534.658.232.452.281 | 61 |
87 | 6243 |
1.193.375.854.875.511.918.605.120.376.079 | 31 |
1.424.145.930.999.858.883.871.649.154.771.774.519.461.783.888.589.990.395.414.241 | 61 |
86 | 6242 |
1.187.054.882.257.789.974.300.764.887.929 | 31 |
1.409.099.293.492.055.619.115.210.663.950.593.660.125.119.165.502.943.929.909.041 | 61 |
85 | 5267 |
228.106.257.936.763.286.220.319.446.695 | 30 |
52.032.464.909.913.183.706.998.947.070.107.074.989.960.738.131.990.946.423.025 | 59 |
84 | 5265 |
130.303.161.860.097.498.924.329.036.469 | 30 |
16.978.913.990.738.767.495.836.054.167.876.145.063.859.476.783.709.931.987.961 | 59 |
83 | 5172 |
Prime! 102.956.548.239.095.784.750.974.549.051 | 30 |
10.600.050.825.309.257.335.789.552.697.979.625.598.753.375.290.352.805.000.601 | 59 |
82 | 4790 |
13.023.229.862.595.863.669.058.813.031 | 29 |
169.604.516.054.008.678.101.490.292.111.292.094.101.876.800.450.615.406.961 | 57 |
81 | 4789 |
11.219.988.402.958.206.841.478.502.211 | 29 |
125.888.139.762.516.652.901.129.855.030.558.921.109.256.615.267.931.888.521 | 57 |
80 | 3994 |
2.589.537.706.161.357.166.603.750.274 | 28 |
6.705.705.531.631.423.370.133.506.726.276.053.310.733.241.361.355.075.076 | 55 |
79 | 3992 |
1.373.512.530.649.258.635.292.477.609 | 28 |
1.886.536.671.850.530.641.991.373.196.913.731.991.460.350.581.766.356.881 | 55 |
78 | 3991 |
1.208.997.217.707.381.836.451.672.371 | 28 |
1.461.674.272.424.190.432.753.231.165.611.323.572.340.914.242.724.761.641 | 55 |
77 | 3978 |
1.103.866.360.472.916.946.437.760.989 | 28 |
1.218.520.941.783.723.816.716.782.817.182.876.176.183.273.871.490.258.121 | 55 |
76 | 3611 |
307.319.174.379.604.964.632.119.143 | 27 |
94.445.074.941.362.044.498.184.066.966.048.189.444.026.314.947.054.449 | 53 |
75 | 3610 |
259.534.999.916.657.357.224.172.376 | 27 |
67.358.416.181.739.334.421.297.151.315.179.212.443.393.718.161.485.376 | 53 |
74 | 3609 |
219.293.240.651.172.832.756.867.922 | 27 |
48.089.525.395.293.201.014.489.455.855.498.441.010.239.259.352.598.084 | 53 |
73 | 3606 |
122.063.831.551.139.898.460.740.721 | 27 |
14.899.578.972.945.056.149.893.218.681.239.894.165.054.927.987.599.841 | 53 |
72 | 3605 |
121.096.650.591.136.334.960.853.879 | 27 |
14.664.398.784.391.760.063.516.738.283.761.536.006.719.348.789.346.641 | 53 |
71 | 2974 |
22.859.776.159.292.529.974.209.485 | 26 |
522.569.366.052.959.132.741.539.313.935.147.231.959.250.663.965.225 | 51 |
70 | 2672 |
2.198.834.453.254.769.178.871.872 | 25 |
4.834.872.952.820.199.705.196.890.986.915.079.910.282.592.784.384 | 49 |
69 | 2671 |
2.107.561.207.005.423.534.632.462 | 25 |
4.441.814.241.274.157.711.392.425.242.931.177.514.721.424.181.444 | 49 |
68 | 2173 |
136.569.391.518.194.940.977.041 | 24 |
18.651.198.699.650.016.285.556.865.558.261.005.699.689.115.681 | 47 |
67 | 1940 |
25.886.574.803.023.261.617.726 | 23 |
670.114.755.032.518.816.023.606.320.618.815.230.557.411.076 | 45 |
66 | 1939 |
25.145.697.104.636.357.696.394 | 23 |
632.306.082.878.117.302.583.050.385.203.711.878.280.603.236 | 45 |
65 | 1938 |
20.706.193.716.777.033.766.268 | 23 |
428.746.458.236.696.712.033.040.330.217.696.632.854.647.824 | 45 |
64 | 1935 |
12.093.315.807.217.013.334.121 | 23 |
146.248.287.213.084.882.816.757.618.288.480.312.782.842.641 | 45 |
63 | 1555 |
3.036.233.455.854.775.865.623 | 22 |
9.218.713.598.451.835.185.192.915.815.381.548.953.178.129 | 43 |
62 | 1554 |
2.211.007.906.320.264.673.878 | 22 |
4.888.555.961.810.720.288.016.108.820.270.181.695.558.884 | 43 |
61 | 1553 |
2.016.192.509.426.760.663.598 | 22 |
4.065.032.235.068.578.387.346.437.838.758.605.322.305.604 | 43 |
60 | 1551 |
1.121.621.184.033.823.231.539 | 22 |
1.258.034.080.473.435.562.012.102.655.343.740.804.308.521 | 43 |
59 | 1373 |
263.397.138.410.176.890.086 | 21 |
69.378.052.522.669.882.013.031.028.896.622.525.087.396 | 41 |
58 | 1372 |
260.088.804.053.044.407.026 | 21 |
67.646.185.993.742.928.767.376.782.924.739.958.164.676 | 41 |
57 | 1371 |
229.359.782.235.085.482.225 | 21 |
52.605.909.706.925.833.964.246.933.852.960.790.950.625 | 41 |
56 | 1368 |
128.501.150.236.577.373.669 | 21 |
16.512.545.612.123.429.216.861.292.432.121.654.521.561 | 41 |
55 | 1082 |
25.686.162.978.506.292.366 | 20 |
659.778.968.558.387.244.939.442.783.855.869.877.956 | 39 |
54 | 950 |
2.282.211.769.458.230.805 | 19 |
5.208.490.560.653.668.833.388.663.560.650.948.025 | 37 |
53 | 947 |
1.363.859.210.193.543.291 | 19 |
1.860.111.945.229.755.699.965.579.225.491.110.681 | 37 |
52 | 738 |
306.950.094.269.977.057 | 18 |
94.218.360.372.347.802.120.874.327.306.381.249 | 35 |
51 | 737 |
228.138.929.476.341.405 | 18 |
52.047.371.142.611.077.177.011.624.117.374.025 | 35 |
50 | 736 |
207.254.460.945.174.618 | 18 |
42.954.411.581.674.911.011.947.618.511.445.924 | 35 |
49 | 734 |
135.772.344.267.730.059 | 18 |
18.434.129.467.955.011.411.055.976.492.143.481 | 35 |
48 | 637 |
13.661.181.333.262.459 | 17 |
186.627.875.420.278.656.872.024.578.726.681 | 33 |
47 | 636 |
13.593.470.459.544.309 | 17 |
184.782.439.134.503.767.305.431.934.287.481 | 33 |
46 | 635 |
12.797.593.520.483.481 | 17 |
163.778.399.915.520.777.025.519.993.877.361 | 33 |
45 | 634 |
11.863.792.420.598.929 | 17 |
140.749.570.599.060.595.060.995.075.947.041 | 33 |
44 | 633 |
11.843.191.515.764.821 | 17 |
140.261.185.279.083.838.380.972.581.162.041 | 33 |
43 | 485 |
3.138.199.296.186.067 | 16 |
9.848.294.822.582.726.272.852.284.928.489 | 31 |
42 | 484 |
3.107.974.295.870.663 | 16 |
9.659.504.223.792.743.472.973.224.059.569 | 31 |
41 | 483 |
3.066.446.727.654.243 | 16 |
9.403.095.533.541.415.141.453.355.903.049 | 31 |
40 | 482 |
2.564.053.868.197.734 | 16 |
6.574.372.239.019.762.679.109.322.734.756 | 31 |
39 | 481 |
2.201.019.508.986.478 | 16 |
4.844.486.878.939.076.709.398.786.844.484 | 31 |
38 | 415 |
314.155.324.482.867 | 15 |
98.693.567.900.935.453.900.976.539.689 | 29 |
37 | 414 |
210.786.628.549.538 | 15 |
44.431.002.775.280.908.257.720.013.444 | 29 |
36 | 411 |
129.610.990.752.569 | 15 |
16.799.008.923.862.526.832.980.099.761 | 29 |
35 | 308 |
30.395.080.190.573 | 14 |
923.860.899.791.363.197.998.068.329 | 27 |
34 | 263 |
3.069.306.930.693 | 13 |
9.420.645.034.800.084.305.460.249 | 25 |
33 | 262 |
2.634.812.417.864 | 13 |
6.942.236.477.330.337.746.322.496 | 25 |
32 | 261 |
2.149.099.165.358 | 13 |
4.618.627.222.542.452.227.268.164 | 25 |
31 | 258 |
1.349.465.117.841 | 13 |
1.821.056.104.269.624.016.501.281 | 25 |
30 | 192 |
128.817.084.669 | 12 |
16.593.841.302.620.314.839.561 | 23 |
29 | 191 |
112.247.658.961 | 12 |
12.599.536.942.224.963.599.521 | 23 |
28 | 190 |
111.283.619.361 | 12 |
12.384.043.938.083.934.048.321 | 23 |
27 | 178 |
101.116.809.851 | 12 |
10.224.609.234.443.290.642.201 | 23 |
26 | 161 |
30.693.069.307 | 11 |
942.064.503.484.305.460.249 | 21 |
25 | 160 |
30.101.273.647 | 11 |
906.086.675.171.576.680.609 | 21 |
24 | 159 |
22.865.150.135 | 11 |
522.815.090.696.090.518.225 | 21 |
23 | 156 |
13.579.355.059 | 11 |
184.398.883.818.388.893.481 | 21 |
22 | 137 |
10.207.355.549 | 11 |
104.190.107.303.701.091.401 | 21 |
21 | 130 |
10.106.064.399 | 11 |
102.132.537.636.735.231.201 | 21 |
20 | 113 |
2.481.623.254 | 10 |
6.158.453.974.793.548.516 | 19 |
19 | 112 |
2.062.386.218 | 10 |
4.253.436.912.196.343.524 | 19 |
18 | 95 |
306.930.693 | 9 |
94.206.450.305.460.249 | 17 |
17 | 69 |
30.001.253 | 8 |
900.075.181.570.009 | 15 |
16 | 67 |
12.866.669 | 8 |
165.551.171.155.561 | 15 |
15 | 66 |
12.028.229 | 8 |
144.678.292.876.441 | 15 |
14 | 65 |
11.129.361 | 8 |
123.862.676.268.321 | 15 |
13 | 56 |
3.069.307 | 7 |
9.420.645.460.249 | 13 |
12 | 55 |
2.294.675 | 7 |
5.265.533.355.625 | 13 |
11 | 54 |
2.012.748 | 7 |
4.051.154.511.504 | 13 |
10 | 51 |
1.270.869 | 7 |
1.615.108.015.161 | 13 |
9 | 44 |
1.042.151 | 7 |
1.086.078.706.801 | 13 |
8 | 31 |
30.693 | 5 |
942.060.249 | 9 |
7 | 30 |
24.846 | 5 |
617.323.716 | 9 |
6 | 29 |
22.865 | 5 |
522.808.225 | 9 |
5 | 20 |
2.636 | 4 |
6.948.496 | 7 |
4 | 19 |
2.285 | 4 |
5.221.225 | 7 |
3 | 14 |
307 | 3 |
94.249 | 5 |
2 | 13 |
264 | 3 |
69.696 | 5 |
1 | 7 |
26 | 2 |
676 | 3 |
Ultra Square Palindromes
| → 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 |
Basenumber | Length |
Ultra Square Palindromes | Length |
| | |
By David Griffeath who conjectures that there are only 21 USP's (see PDF article) |
21 | 741 |
795.559.265.009.384.106 | 18 |
632.914.544.142.271.449.944.172.241.445.419.236 | 36 |
20 | 628 |
11.100.100.100.100.111 | 17 |
123.212.222.232.242.494.242.232.222.212.321 | 33 |
19 | 410 |
111.100.010.001.111 | 15 |
12.343.212.222.246.964.222.221.234.321 | 29 |
18 | 408 |
111.010.010.010.111 | 15 |
12.323.222.322.444.944.422.322.232.321 | 29 |
17 | 406 |
111.001.010.100.111 | 15 |
12.321.224.243.244.944.234.242.212.321 | 29 |
16 | 397 |
110.101.010.101.011 | 15 |
12.122.232.425.262.926.252.423.222.121 | 29 |
15 | 257 |
1.111.001.001.111 | 13 |
1.234.323.224.469.644.223.234.321 | 25 |
14 | 255 |
1.110.101.010.111 | 13 |
1.232.324.252.649.462.524.232.321 | 25 |
13 | 253 |
1.110.011.100.111 | 13 |
1.232.124.642.369.632.464.212.321 | 25 |
12 | 248 |
1.101.101.011.011 | 13 |
1.212.423.436.449.446.343.242.121 | 25 |
11 | 246 |
1.101.011.101.011 | 13 |
1.212.225.444.549.454.445.222.121 | 25 |
10 | 156 |
13.579.355.059 | 11 |
184.398.883.818.388.893.481 | 21 |
9 | 155 |
11.110.101.111 | 11 |
123.434.346.696.643.434.321 | 21 |
8 | 153 |
11.101.110.111 | 11 |
123.234.645.696.546.432.321 | 21 |
7 | 148 |
11.011.111.011 | 11 |
121.244.565.696.565.442.121 | 21 |
6 | 113 |
2.481.623.254 | 10 |
6.158.453.974.793.548.516 | 19 |
5 | 92 |
111.111.111 | 9 |
12.345.678.987.654.321 | 17 |
4 | 90 |
111.091.111 | 9 |
12.341.234.943.214.321 | 17 |
3 | 24 |
11.011 | 5 |
1.212.42.121 | 10 |
2 | 17 |
1.111 | 4 |
1.234.321 | 7 |
1 | 12 |
212 | 3 |
44.944 | 5 |
Non Sporadic Square Palindromes → a Pari/gp script generator
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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