Consider the number 28235 that is my diurnal age today. It has a circulant matrix with a determinant 10100 that is a pronic number since 10100 = 100 x 101.$$\begin{bmatrix}
2 & 8 & 2 & 3 & 5 \\
8 & 2 & 3 & 5 & 2 \\
2 & 3 & 5 & 2 & 8 \\
3 & 5 & 2 & 8 & 2 \\
5 & 2 & 8 & 2 & 3
\end{bmatrix}$$What's interesting is that most of the permutations of the digits of 28235 have determinants of their circulant matrices that are also pronic (permalink).
Number | Determinant | Factorisation ------------------------------------------------ 22358 | 15500 | 124 x 125 22385 | 10100 | 100 x 101 22538 | 19100 | 22583 | 10100 | 100 x 101 22835 | 19100 | 22853 | 15500 | 124 x 125 23258 | 19100 | 23285 | 19100 | 23528 | 10100 | 100 x 101 23582 | 15500 | 124 x 125 23825 | 15500 | 124 x 125 23852 | 10100 | 100 x 101 25238 | 15500 | 124 x 125 25283 | 15500 | 124 x 125 25328 | 10100 | 100 x 101 25382 | 19100 | 25823 | 19100 | 25832 | 10100 | 100 x 101 28235 | 10100 | 100 x 101 28253 | 10100 | 100 x 101 28325 | 15500 | 124 x 125 28352 | 19100 | 28523 | 19100 | 28532 | 15500 | 124 x 125 32258 | 10100 | 100 x 101 32285 | 15500 | 124 x 125 32528 | 15500 | 124 x 125 32582 | 19100 | 32825 | 10100 | 100 x 101 32852 | 19100 | 35228 | 19100 | 35282 | 10100 | 100 x 101 35822 | 15500 | 124 x 125 38225 | 19100 | 38252 | 15500 | 124 x 125 38522 | 10100 | 100 x 101 52238 | 10100 | 100 x 101 52283 | 19100 | 52328 | 19100 | 52382 | 15500 | 124 x 125 52823 | 10100 | 100 x 101 52832 | 15500 | 124 x 125 53228 | 15500 | 124 x 125 53282 | 10100 | 100 x 101 53822 | 19100 | 58223 | 15500 | 124 x 125 58232 | 19100 | 58322 | 10100 | 100 x 101 82235 | 15500 | 124 x 125 82253 | 19100 | 82325 | 19100 | 82352 | 10100 | 100 x 101 82523 | 15500 | 124 x 125 82532 | 10100 | 100 x 101 83225 | 10100 | 100 x 101 83252 | 15500 | 124 x 125 83522 | 19100 | 85223 | 10100 | 100 x 101 85232 | 19100 | 85322 | 15500 | 124 x 125
Note that it is only when the determinant is 19100 that it is not pronic since 19100 = 100 x 191. In my post titled Determinants of Circulant Matrices, I listed all numbers up to 40000 with the property that the determinants of their circulant matrices were pronic. The numbers between 28000 and 40000 are:
28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995
Note that with 28235 and its digit permutations there are TWO determinants that satisfy. These are:
- \(10100 = 100 \times 101\)
- \(15500 = 124 \times 125\)
Number | Determinant | Factorisation ------------------------------------------------ 22378 | 14762 | 121 x 122 22387 | 11462 | 22738 | 27962 | 22783 | 11462 | 22837 | 27962 | 22873 | 14762 | 121 x 122 23278 | 27962 | 23287 | 27962 | 23728 | 11462 | 23782 | 14762 | 121 x 122 23827 | 14762 | 121 x 122 23872 | 11462 | 27238 | 14762 | 121 x 122 27283 | 14762 | 121 x 122 27328 | 11462 | 27382 | 27962 | 27823 | 27962 | 27832 | 11462 | 28237 | 11462 | 28273 | 11462 | 28327 | 14762 | 121 x 122 28372 | 27962 | 28723 | 27962 | 28732 | 14762 | 121 x 122 32278 | 11462 | 32287 | 14762 | 121 x 122 32728 | 14762 | 121 x 122 32782 | 27962 | 32827 | 11462 | 32872 | 27962 | 37228 | 27962 | 37282 | 11462 | 37822 | 14762 | 121 x 122 38227 | 27962 | 38272 | 14762 | 121 x 122 38722 | 11462 | 72238 | 11462 | 72283 | 27962 | 72328 | 27962 | 72382 | 14762 | 121 x 122 72823 | 11462 | 72832 | 14762 | 121 x 122 73228 | 14762 | 121 x 122 73282 | 11462 | 73822 | 27962 | 78223 | 14762 | 121 x 122 78232 | 27962 | 78322 | 11462 | 82237 | 14762 | 121 x 122 82273 | 27962 | 82327 | 27962 | 82372 | 11462 | 82723 | 14762 | 121 x 122 82732 | 11462 | 83227 | 11462 | 83272 | 14762 | 121 x 122 83722 | 27962 | 87223 | 11462 | 87232 | 27962 | 87322 | 14762 | 121 x 122
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