Maillet's determinant
In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by (Maillet 1913) whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p (Muir 1930). (Malo 1914) calculated the determinant Dp for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (–p)(p – 3)/2, and conjectured that it is given by this formula in general. (Carlitz Olson) showed that this conjecture is incorrect; the determinant in general is given by Dp = (–p)(p – 3)/2h−, where h− is the first factor of the class number of the cyclotomic field generated by pth roots of 1, which happens to be 1 for p less than 23. In particular, this verifies Maillet's conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it. Their results have been extended to all non-prime odd numbers by K. Wang(1982).
References
- Carlitz, L.; Olson, F. R. (1955), "Maillet's determinant", Proceedings of the American Mathematical Society 6 (2): 265–269, doi:10.2307/2032352, ISSN 0002-9939
- Maillet, E. (1913), "Question 4269", L'Intermédiaire des Mathématiciens xx: 218, https://books.google.com/books?id=0ABSAQAAIAAJ
- Malo, E. (1914), "Sur un certain déterminant d'ordre premier", L'Intermédiaire des Mathématiciens xxi: 173–176, http://babel.hathitrust.org/cgi/pt?id=pst.000052363436;view=1up;seq=189
- Muir, Thomas (1930), Contributions To The History Of Determinants 1900–1920, Blackie And Son Limited., https://archive.org/details/contributionstot032405mbp
- Wang, Kai (1984), "On Maillet determinant", Journal of Number Theory (Journal of Number Theory 18) 18 (3): 306–312, doi:10.1016/0022-314X(84)90064-7
 |  |
