A228615 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as 2*(i + j) - 1 and 2*(i + j) + 1 are twin primes or not.

1, -1, -1, -1, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, -1, -1, 1, 1, 1, -1, 2, 8, -18, -9, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4096, -4096, 64, -20, -125, 5, -6, -216, 24, 152, 54872, -106742, 14045, 125, -21125, -274625, -274625, 10985, -16731, -970299, 1804275, 1312200, 373248, -691488, -192080

Offset: 1

Author

Zhi-Wei Sun, Aug 27 2013

Keywords

• sign

Comments

Conjecture: a(n) is nonzero for any n > 35.

Clearly this conjecture implies the twin prime conjecture.

Examples

			a(1) = 1 since 2*(1 + 1) - 1 = 3 and 2*(1 + 1) + 1 = 5 are twin primes.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..500 (First 200 terms from Zhi-Wei Sun)

Crossrefs

Cf. A001359, A006512, A069191, A228557, A228591.

Programs

Mathematica

a[n_]:=a[n]=Det[Table[If[PrimeQ[2(i+j)-1]&&PrimeQ[2(i+j)+1],1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,20}]
Table[Det[Table[If[AllTrue[2(i+j)+{1,-1},PrimeQ],1,0],{i,k},{j,k}]],{k,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2019 *)