A228623 Determinant of the n X n matrix with (i,j)-entry (i,j = 0,...,n-1) equal to 1 or 0 according as n + i - j and n - i + j are both prime or not.

0, 1, 1, 1, 0, -1, 0, 0, 0, -4, -1, 0, 0, 0, -6, 0, 0, -144, 0, 0, 0, -1, 168, 1024, 420, 0, 0, 0, -1, -9801, 0, 144, 0, 0, 3072, 7056, 0, 0, -42346434, 0, 0, -331776, 0, 0, 36528128, -104976, 96545145, 0, 34665386, -62500, 2826240, 2025, 0, -23174596, 0, 0, 255578880, -4, -3, 990172089

Offset: 1

Author

Zhi-Wei Sun, Aug 27 2013

Keywords

• sign

Comments

Conjecture: a(n) is nonzero if n is odd and greater than 120.

This implies Goldbach's conjecture for even numbers of the form 4*k + 2.

Examples

			a(1) = 0 since 1 + 0 - 0 = 1 is not a prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A002372, A228591, A228615, A228616, A228557, A228559.

Programs

Mathematica

a[n_]:=Det[Table[If[PrimeQ[n+j-i]==True&&PrimeQ[n+i-j]==True,1,0],{i,0,n-1},{j,0,n-1}]]
Table[a[n],{n,1,20}]