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 : 7, 3113, 44611644
Pentagonal numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only
base x ( 3 x base - 1 ) |
------------------------------- |
2 |
|
Normal and Palindromic Pentagonals
So far I compiled 76 Palindromic Pentagonals.
Here is the largest Sporadic Palindromic Pentagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ]
This basenumber 829.412.616.490.882.316.063.874.786
has 27 digits
yielding the following palindromic pentagonal number
1.031.887.932.591.377.142.105.479.799.979.745.012.417.731.952.397.881.301
with a length of 55 digits.
|
All palindromic pentagonal numbers can only end with 0, 1, 2, 5, 6 or 7.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by 0 or 5 : 10 or 15
2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by 2 or 7 : 62 or 67
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
There exist no palindromic pentagonals of length 3, 9, 11, 12, 24, 30, 32, 33, 42, 45, 46, 50, 51, 52, 54.
(Sloane's A059868)
Case Penta | Change of variables | CUDApalin parameters | Base Correction |
odd base | n = 2 * m + 1 | A B C → 6 5 1 | CUDAbase * 2 + 1 |
even base | n = 2 * m + 2 | A B C → 6 11 5 | CUDAbase * 2 + 2 |
Believe it or not but some numbers can't be expressed as the sum of 3 pentagonal numbers.
%N Not the sum of 3 pentagonal numbers. under Sloane's A003679.
4, 8, 9, 16, 19, ...
Every pentagonal number is one-third of a triangular number.

A good source for such statements is “The Book of Numbers”
by John H. Conway and Richard K. Guy.
Click on the image on the left for more background about the book.
The book can be ordered at 'www.amazon.com'.
Sloane's A014979 gives the first numbers that are both Pentagonal and Triangular.
(Sequence corrected and extended by Warut Roonguthai.)
1, 210, 40755, 7906276, 1533776805, 297544793910, ...
Consult also Eric Weisstein's page Pentagonal Triangular Number.
Sloane's A036353 gives the first numbers that are both Pentagonal and Square.
1, 9801, 94109401, 903638458801, 8676736387298001, ...
Consult also Eric Weisstein's page Pentagonal Square Number.
Sloane's A046180 gives the first numbers that are both Pentagonal and Hexagonal.
1, 40755, 1533776805, 57722156241751, ...
Consult also Eric Weisstein's page Pentagonal Hexagonal Number.
Sloane's A048900 gives the first numbers that are both Pentagonal and Heptagonal.
1, 4347, 16701685, 246532939589097, ...
Consult also Eric Weisstein's page Pentagonal Heptagonal Number.
Sloane's A046189 gives the first numbers that are both Pentagonal and Octagonal.
1, 176, 1575425, 234631320, 2098015778145, ...
Consult also Eric Weisstein's page Pentagonal Octagonal Number.
Sloane's A048915 gives the first numbers that are both Pentagonal and Nonagonal.
1, 651, 180868051, 95317119801, 26472137730696901, ...
Consult also Eric Weisstein's page Pentagonal Nonagonal Number.
The best way to get a 'structural' insight as how to imagine pentagonals is to visit for instance this site :
Sources Revealed
Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular pentagonal numbers at
%N Pentagonal numbers n(3n1)/2 under A000326.
The palindromic pentagonal numbers are categorised as follows :
%N n(3n1)/2 is a pentagonal palindrome under A028386.
%N Pentagonal palindromes under A002069.
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.
|
The Table
Index Nr | Info | Basenumber | Length |
Palindromic Pentagonals | Length |
| | |
[PG5] Formula = n(3n1)/2
|
76 | Info | 829.412.616.490.882.316.063.874.786 | 27 |
1.031.887.932.591.377.142.105.479.799.979.745.012.417.731.952.397.881.301 | 55 |
75 | Info | 219.838.383.896.945.839.241.529.093 | 27 |
72.493.372.551.631.403.519.074.864.446.847.091.530.413.615.527.339.427 | 53 |
74 | Info | 211.567.096.617.431.127.327.768.331 | 27 |
67.140.954.556.694.156.338.058.353.235.385.083.365.149.665.545.904.176 | 53 |
73 | Info | 196.744.195.862.034.221.724.924.905 | 27 |
58.062.417.908.097.725.661.820.084.648.002.816.652.779.080.971.426.085 | 53 |
72 | Info | 140.815.521.353.961.270.534.020.208 | 27 |
29.743.516.581.281.883.860.142.882.928.824.106.838.818.218.561.534.792 | 53 |
71 | Info | 123.470.453.704.578.756.803.923.204 | 27 |
22.867.429.407.021.789.074.918.527.272.581.947.098.712.070.492.476.822 | 53 |
70 | Info | 115.716.935.699.622.852.551.776.543 | 27 |
20.085.613.811.565.974.693.953.734.743.735.939.647.956.511.831.658.002 | 53 |
69 | Info | 1.997.352.210.935.459.325.350.510 | 25 |
5.984.123.781.793.151.403.874.475.744.783.041.513.971.873.214.895 | 49 |
68 | Info | 1.366.335.687.983.609.354.348.844 | 25 |
2.800.309.818.386.464.643.692.792.972.963.464.646.838.189.030.082 | 49 |
67 | Info | 1.214.693.643.268.411.793.685.388 | 25 |
2.213.220.970.495.031.472.095.671.765.902.741.305.940.790.223.122 | 49 |
66 | Info | 646.108.029.840.746.975.319.151 | 24 |
626.183.379.337.037.375.893.787.787.398.573.730.733.973.381.626 | 48 |
65 | Info | 386.808.141.157.899.128.841.063 | 24 |
224.430.807.099.043.827.014.991.199.410.728.340.990.708.034.422 | 48 |
64 | Info | 128.481.111.918.761.748.263.908 | 24 |
24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.742 | 47 |
63 | Info | 4.181.240.246.129.576.678.503 | 22 |
26.224.154.993.780.584.443.322.334.448.508.739.945.142.262 | 44 |
62 | Info | 3.166.575.018.238.865.276.806 | 22 |
15.040.796.019.201.704.940.688.604.940.710.291.069.704.051 | 44 |
61 | Info | 1.161.338.314.703.013.743.743 | 22 |
2.023.060.021.795.854.282.372.732.824.585.971.200.603.202 | 43 |
60 | Info | 226.255.113.072.008.674.109 | 21 |
76.787.064.286.841.145.600.200.654.114.868.246.078.767 | 41 |
59 | Info | 101.908.013.398.232.552.381 | 21 |
15.577.864.792.161.518.113.031.181.516.129.746.877.551 | 41 |
58 | Info | 32.178.927.183.122.554.461 | 20 |
1.553.225.031.985.055.486.446.845.505.891.305.223.551 | 40 |
57 | Info | 12.470.286.104.989.637.219 | 20 |
233.262.053.310.446.425.999.524.644.013.350.262.332 | 39 |
56 | Info | 3.225.069.471.604.675.446 | 19 |
15.601.609.645.014.690.722.709.641.054.690.610.651 | 38 |
55 | Info | 2.215.732.656.045.463.378 | 19 |
7.364.206.804.599.425.576.755.249.954.086.024.637 | 37 |
54 | Info | 642.912.991.153.879.996 | 18 |
620.005.671.291.643.466.664.346.192.176.500.026 | 36 |
53 | Info | 222.497.695.855.916.654 | 18 |
74.257.836.991.787.986.368.978.719.963.875.247 | 35 |
52 | Info | 64.541.848.291.142.556 | 17 |
6.248.475.271.255.291.881.925.521.725.748.426 | 34 |
51 | Info | 1.343.985.615.421.963 | 16 |
2.709.446.001.691.728.271.961.006.449.072 | 31 |
50 | Info | 1.331.901.641.608.903 | 16 |
2.660.942.974.380.735.370.834.792.490.662 | 31 |
49 | Info | 192.460.705.375.190 | 15 |
55.561.684.670.273.437.207.648.616.555 | 29 |
48 | Info | 189.271.045.290.605 | 15 |
53.735.292.878.097.279.087.829.253.735 | 29 |
47 | Info | 62.640.018.209.362 | 14 |
5.885.657.821.903.773.091.287.565.885 | 28 |
46 | Info | 62.407.875.352.937 | 14 |
5.842.114.359.101.551.019.534.112.485 | 28 |
45 | Info | 22.073.207.482.889 | 14 |
730.839.732.873.989.378.237.938.037 | 27 |
44 | Info | 8.505.462.902.906 | 13 |
108.514.348.789.060.987.843.415.801 | 27 |
43 | Info | 6.106.091.136.277 | 13 |
55.926.523.446.777.764.432.562.955 | 26 |
42 | Info | 3.690.877.000.943 | 13 |
20.433.859.554.133.145.595.833.402 | 26 |
41 | Info | 2.704.211.220.546 | 13 |
10.969.137.487.988.978.473.196.901 | 26 |
40 | Info | 2.269.269.065.474 | 13 |
7.724.373.137.274.727.313.734.277 | 25 |
39 | Info | 2.118.668.671.851 | 13 |
6.733.135.411.623.261.145.313.376 | 25 |
38 | Info | 1.199.186.171.664 | 13 |
2.157.071.211.464.641.121.707.512 | 25 |
37 | Info | 100.208.264.581 | 12 |
15.062.544.435.453.444.526.051 | 23 |
36 | Info | 72.507.025.789 | 11 |
7.885.903.183.113.813.095.887 | 22 |
35 | Info | 20.483.169.156 | 11 |
629.340.327.999.723.043.926 | 21 |
34 | Info | 18.517.422.797 | 11 |
514.342.420.555.024.243.415 | 21 |
33 | Info | 18.306.458.265 | 11 |
502.689.621.303.126.986.205 | 21 |
32 | Info | 6.693.818.696 | 10 |
67.210.813.099.031.801.276 | 20 |
31 | Info | 2.049.123.156 | 10 |
6.298.358.561.658.538.926 | 19 |
30 | Info | 1.345.679.688 | 10 |
2.716.280.733.370.826.172 | 19 |
29 | Info | 851.649.306 | 9 |
1.087.959.810.189.597.801 | 19 |
28 | Info | 818.049.201 | 9 |
1.003.806.742.476.083.001 | 19 |
27 | Info | 700.457.153 | 9 |
735.960.334.433.069.537 | 18 |
26 | Info | 316.882.086 | 9 |
150.621.384.483.126.051 | 18 |
25 | Info | 225.563.533 | 9 |
76.318.361.016.381.367 | 17 |
24 | Info | 100.058.581 | 9 |
15.017.579.397.571.051 | 17 |
23 | Info | 84.885.761 | 8 |
10.808.388.588.380.801 | 17 |
22 | Info | 59.401.650 | 8 |
5.292.834.004.382.925 | 16 |
21 | Info | 13.280.839 | 8 |
264.571.020.175.462 | 15 |
20 | Info | 2.684.561 | 7 |
10.810.300.301.801 | 14 |
19 | Info | 1.979.130 | 7 |
5.875.432.345.785 | 13 |
18 | Info | 1.965.117 | 7 |
5.792.526.252.975 | 13 |
17 | Info |
Prime Curios! 1.258.723 | 7 |
2.376.574.756.732 | 13 |
16 | Info | Prime! 44.059 | 5 |
2.911.771.192 | 10 |
15 | Info | 31.926 | 5 |
1.528.888.251 | 10 |
14 | Info | 26.906 | 5 |
1.085.885.801 | 10 |
13 | Info | 26.466 | 5 |
1.050.660.501 | 10 |
12 | Info | 6.010 | 4 |
54.177.145 | 8 |
11 | Info | 4.228 | 4 |
26.811.862 | 8 |
10 | Info | 2.229 | 4 |
7.451.547 | 7 |
9 | Info | 2.173 | 4 |
7.081.807 | 7 |
8 | Info | 693 | 3 |
720.027 | 6 |
7 | Info | Prime! 101 | 3 |
15.251 | 5 |
6 | Info | 44 | 2 |
2.882 | 4 |
5 | Info | 26 | 2 |
1.001 | 4 |
4 | Info | 4 | 1 |
22 | 2 |
3 | Info | Prime! 2 | 1 |
Prime! 5 | 1 |
2 | Info | 1 | 1 |
1 | 1 |
1 | Info | 0 | 1 |
0 | 1 |
Contributions
Exhaustive search performed up to length 29 by Patrick De Geest.
Higher lengths up to length 47 were found and submitted by Feng Yuan.
Warut Roonguthai (email) from Thailand helped searching for these pentagonal palindromes.
He discovered those starting from index number [26] up to [36].
Feng Yuan (email) from Washington State, USA, discovered the palindromic pentagonals
starting from index number [50] and [51].
[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, submitted the palindromic pentagonals
starting from index number [52] up to [59].
[ November 4, 2022 ]
Robert Xiao (email) added three missing entries [55], [56] and [57].
[ November 11, 2022 ]
Robert Xiao (email) added two missing entries [60] and [62].
[ November 11, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) added two entries [65] and [66].
[ December 6, 2022 ]
Robert Xiao (email) added three new record entries [67], [68] and [69].
[ January 9, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic pentagonals
just beyond the cutoff of Robert's global search up to length 52. In total six new entries [70] up to [75] were added.
[ June 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [76].
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