Circulant Matrix to the Rescue

I was stuck for several days on a number associated with my diurnal age: 27164. I couldn't anything of interest about this number, or at least nothing that interested me. In the end, I remembered the circulant matrix that is associated with each number. For the case of 27164 this matrix is shown below. $$ \begin{pmatrix}2&7&1&6&4\\4&2&7&1&6\\6&4&2&7&1\\1&6&4&2&7\\7&1&6&4&2 \end{pmatrix}$$The determinant of this matrix is 100 and there are only 89 numbers in the range up to one million that have 100 as their determinant. These numbers are:

799, 979, 997, 12674, 14762, 16427, 17246, 21476, 23564, 24617, 24653, 25436, 26345, 26741, 27164, 32465, 33455, 33554, 34445, 34454, 34526, 34535, 34544, 34553, 35345, 35354, 35435, 35444, 35543, 35642, 36254, 41267, 42356, 42716, 43355, 43445, 43454, 43535, 43544, 43625, 44345, 44354, 44435, 44453, 44534, 44543, 45263, 45344, 45353, 45434, 45443, 45533, 46172, 46532, 47621, 52634, 53246, 53345, 53444, 53453, 53534, 53543, 54335, 54344, 54353, 54362, 54434, 54443, 55334, 55433, 56423, 61724, 62147, 62543, 63452, 64235, 64271, 65324, 67412, 71642, 72461, 74126, 76214, 303040, 304030, 403030, 889898, 988898, 989888

Of these, many are permutations of the digits of 27164. These permutations are shown below in bold with 27164 itself marked in red.

12467, 12476, 12647, 12674, 12746, 12764, 14267, 14276, 14627, 14672, 14726, 14762, 16247, 16274, 16427, 16472, 16724, 16742, 17246, 17264, 17426, 17462, 17624, 17642, 21467, 21476, 21647, 21674, 21746, 21764, 24167, 24176, 24617, 24671, 24716, 24761, 26147, 26174, 26417, 26471, 26714, 26741, 27146, 27164, 27416, 27461, 27614, 27641, 41267, 41276, 41627, 41672, 41726, 41762, 42167, 42176, 42617, 42671, 42716, 42761, 46127, 46172, 46217, 46271, 46712, 46721, 47126, 47162, 47216, 47261, 47612, 47621, 61247, 61274, 61427, 61472, 61724, 61742, 62147, 62174, 62417, 62471, 62714, 62741, 64127, 64172, 64217, 64271, 64712, 64721, 67124, 67142, 67214, 67241, 67412, 67421, 71246, 71264, 71426, 71462, 71624, 71642, 72146, 72164, 72416, 72461, 72614, 72641, 74126, 74162, 74216, 74261, 74612, 74621, 76124, 76142, 76214, 76241, 76412, 76421

Admittedly, focusing on a determinant of 100 is rather arbitrary, but it is a nice round number and it certainly came to my aid in finding an interesting property of 27164. This prompted me to investigate the range of values for the determinants in the natural numbers up to 40,000. It turns out the minimum value of -19683 occurs with 9909 and the maximum value of 205821 occurs with 30990 and 39009. Figure 1 shows a plot of the values of the determinants.

Figure 1: permalink

As can be seen it is four digit numbers from 1000 up to 9999 that fluctuate between positive and negative. The three digit numbers and the five digit numbers up to 40000 are all positive. The positive/negative fluctuations recur once the six digit numbers are reached. This is to be expected of course given the way the determinant is calculated. Figure 2 a close up of the range from 10 to 10000.

Figure 2: permalink