Maillet's determinant

In mathematics, Maillet's determinant D_p 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 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. Carlitz & Olson (1955) 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 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, 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