A228561 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.

1, -1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0, 4, -16, -9, 25, 4, -81, -81, 81, 841, -5929, -3969, 19600, 69169, -667489, -285156, 80656, 276676, -790321, -60025, 3136, 10816, -40000, -45369, 221841, 86436, -168100, -12100, 13225, 11881, -87616, -71289, 729

Offset: 1

Author

Zhi-Wei Sun, Aug 25 2013

Keywords

• sign

Comments

Conjecture: a(n) is nonzero for each n > 28.

This implies that there are infinitely many primes p with 4*p^2 + 1 also prime. Note also that (-1)^{n*(n-1)/2}*a(n) is always a square in view of the comments in A228591.

Examples

			a(1) = 1 since  1 + 1 = 2 and 4*2^2 + 1 = 17 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A052291, A069191, A071524, A228591, A228552, A228557, A228559, A228574, A228578, A228615, A228616.

Programs

Mathematica

a[n_]:=a[n]=Det[Table[If[PrimeQ[i+j]==True&&PrimeQ[4(i+j)^2+1]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]