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, pages 340–342).
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.
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).