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 modulo p (Muir 1930, pages 340–342).
Determinant value 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 (1955) 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 Wang (1984).
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, JSTOR 2032352, MR 0069207 Maillet, E. (1913), "Question 4269", L'Intermédiaire des Mathématiciens, xx: 218 Malo, E. (1914), "Sur un certain déterminant d'ordre premier", L'Intermédiaire des Mathématiciens, xxi: 173–176 Muir, Thomas (1930), Contributions To The History Of Determinants 1900–1920, Blackie And Son Limited. Wang, Kai (1984), "On Maillet determinant", Journal of Number Theory, 18 (3), Journal of Number Theory 18: 306–312, doi:10.1016/0022-314X(84)90064-7