A200522 Least m>0 such that n = 2^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 0, 0, 0, 15, 12, 0, 0, 20, 16, 24, 0, 32, 20, 0, 0, 28, 12, 56, 15, 16, 16, 0, 112, 68, 16, 40, 0, 20, 12, 0, 0, 52, 20, 15, 80, 16, 16, 0, 112, 36, 12, 56, 33, 28, 28, 0, 0, 20, 15, 40, 128, 16, 12, 0, 117, 48, 16, 24, 0, 44, 28, 0, 0, 15, 12, 40, 63

Offset: 0

Author

M. F. Hasler, Nov 18 2011

Keywords

• nonn

Comments

If such an m>0 exists, this proves that n is not in A051213, i.e., not of the form 2^x-y^2. On the other hand, if there are integers x, y such that n=2^x-y^2, then we know that a(n)=0.

a(519) > 20000 if it is nonzero.

It remains to show whether "a(n)=0" is equivalent to "n is in A051213". For example, one can show that 519 is not in A051213, but we don't know a(519) yet. - M. F. Hasler, Oct 23 2014

a(519) = 131235. - Seiichi Azuma, Apr 05 2025

Examples

			See A200507 for motivation and examples.
		

Links

• Seiichi Azuma, Table of n, a(n) for n = 0..1000 (terms up to a(518) from M. F. Hasler)

Crossrefs

Cf. A051204-A051221, A200523, A200524, A200505-A200520.

Programs

PARI

A200522(n,b=2,p=3)={ my( x=0, qr, bx, seen ); for( m=3,9e9, while( x^p < m, issquare(b^x-n) & return(0); x++); qr=vecsort(vector(m,y,y^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}