190929(2)

PARI


A327855

For n > 1, (Sum_{i=0..prime(n)-1} (1+Legendre(i, prime(n)) * x^i)^2 == (-1)^((p - 1)/2) * p mod ((x^p - 1)/(x - 1)) where p is n-th prime

となることを確認しておく。

(16:57) gp > forprime(p=2, 30, print(Vecrev((sum(k=0, p-1, (1+kronecker(k, p))*x^k))^2, 2*p-1), ", "))
[1, 4, 4],
[1, 4, 4, 0, 0],
[1, 4, 4, 0, 4, 8, 0, 0, 4],
[1, 4, 8, 8, 8, 8, 8, 0, 4, 0, 0, 0, 0],
[1, 4, 4, 4, 12, 12, 12, 8, 12, 12, 12, 0, 8, 8, 8, 0, 0, 0, 4, 0, 0],
[1, 4, 4, 4, 12, 8, 4, 8, 4, 4, 12, 8, 12, 24, 8, 8, 8, 0, 4, 8, 4, 8, 8, 0, 4],
[1, 4, 8, 8, 8, 8, 8, 0, 8, 12, 16, 8, 8, 12, 8, 12, 16, 32, 12, 8, 8, 8, 8, 8, 16, 8, 4, 0, 8, 8, 4, 8, 4],
[1, 4, 4, 0, 4, 12, 12, 12, 12, 12, 20, 20, 20, 16, 12, 16, 20, 20, 20, 0, 16, 16, 20, 16, 8, 8, 8, 8, 8, 0, 0, 0, 4, 8, 4, 0, 0],
[1, 4, 8, 12, 16, 16, 16, 16, 16, 20, 24, 16, 24, 20, 24, 24, 24, 24, 24, 24, 24, 24, 24, 0, 20, 16, 12, 8, 8, 8, 8, 8, 4, 0, 8, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0],
[1, 4, 4, 0, 4, 12, 12, 12, 12, 12, 20, 16, 12, 20, 20, 8, 12, 16, 12, 8, 20, 16, 20, 20, 20, 28, 28, 24, 28, 56, 24, 24, 28, 24, 16, 16, 16, 16, 16, 8, 12, 16, 8, 8, 20, 16, 12, 16, 20, 8, 12, 8, 8, 8, 0, 0, 4],
(17:00) gp > forprime(p=2, 30, print(lift(Mod((sum(k=0, p-1, (1+kronecker(k, p))*x^k))^2, polcyclo(p))), ", "))
1,
-3,
5,
-7,
-11,
13,
17,
-19,
-23,
29,
(17:00) gp >