OFFSET
2,5
COMMENTS
Conjecture: a(n) = 0 if and only if p_n == 3 (mod 4).
Note that for an odd prime p we have (((p-1)/2)!)^2 == (-1)^{(p+1)/2} (mod p) by Wilson's theorem. In 1961, Mordell proved that((p-1)/2)! == (-1)^{(h(-p)+1)/2} (mod p) for any prime p > 3 with p == 3 (mod 4), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 2..80
L. J. Mordell, The congruence ((p-1)/2)! == 1 or -1 (mod p), Amer. Math. Monthly 68 (1961), 145-146.
Zhi-Wei Sun, A conjecture on Legendre symbol determinants, a message to Number Theory List, July 17, 2013.
EXAMPLE
a(2) = 0 since the Legendre symbol ((1^2-1)/3) is equal to 0.
MATHEMATICA
a[n_]:=Det[Table[JacobiSymbol[i^2-((Prime[n]-1)/2)!*j, Prime[n]], {i, 1, (Prime[n]-1)/2}, {j, 1, (Prime[n]-1)/2}]]
Table[a[n], {n, 2, 30}]
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Aug 05 2013
STATUS
approved