In mathematics, Maillet's determinant D<sub>p</sub> is the determinant of the matrix introduced by whose entries are R(s/r) for s,r = 1, 2, ..., (p â 1)/2 â Z/pZ for an odd prime p, where and R(a) is the least positive residue of a modulo p .
calculated the determinant D<sub>p</sub> for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (âÂÂp)<sup>(p â 3)/2</sup>, and conjectured that it is given by this formula in general. showed that this conjecture is incorrect; the determinant in general is given by D<sub>p</sub> = (âÂÂp)<sup>(p â 3)/2</sup>h<sup>âÂÂ</sup>, where h<sup>âÂÂ</sup> 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 .