A200507 Least m>0 such that n = 7^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 3, 0, 8, 3, 0, 0, 3, 7, 8, 3, 8, 0, 3, 16, 7, 3, 8, 0, 3, 16, 28, 3, 0, 16, 3, 16, 8, 3, 7, 16, 3, 0, 8, 3, 8, 7, 3, 28, 0, 3, 8, 16, 3, 0, 19, 3, 0, 0, 3, 7, 8, 3, 0, 20, 3, 16, 7, 3, 8, 100, 3, 16, 35, 3, 8, 28, 3, 16, 20, 3, 7, 16, 3, 16, 8, 3, 28

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 A051218, i.e., not of the form 7^x-y^2. On the other hand, if there are integers x, y such that n=7^x-y^2, then we know that a(n)=0.

a(432) is at least of order 10^5.

Examples

			a(2)=3 since quadratic residues mod 3 (i.e. possible values for y^2 mod 3) are {0,1}, and 7^x is always congruent to 1 (mod 3), therefore there cannot be any (x,y) such that 7^x-y^2 = 2. The modulus m=3 is the least number for which this equation has no solution in Z/mZ: For m=1 the equation is always true, and for m=2 one always has the solution x=0 and y=0 (for even n) or y=1 (for odd n).
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..431

Crossrefs

Cf. A051204-A051221, A200505-A200524.

Programs

PARI

A200507(n,b=7,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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}