In mathematics, Maillet's determinant "D""p" is the determinant of the matrix introduced by whose entries are "R"("s/r") for "s","r" = 1, 2, ..., ("p" – 1)/2 ∈ Z/"p"Z for an odd prime "p", where and "R"("a") is the least positive residue of "a" mod "p" . calculated the determinant "D""p" 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. showed that this conjecture is incorrect; the determinant in general is given by "D""p" = (–"p")("p" – 3)/2"h"−, where "h"− is the first factor of the class number of the cyclotomic field generated by "p"th 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).