login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Least m>0 such that n = y^2 - 10^x (mod m) has no solution, or 0 if no such m exists.
13

%I #11 Jan 03 2016 04:37:38

%S 0,3,5,0,3,9,0,3,0,11,3,9,5,3,9,0,3,5,11,3,9,0,3,9,0,3,0,5,3,9,16,3,5,

%T 20,3,0,1001,3,9,0,3,9,5,3,0,56,3,5,0,3,9,11,3,11,0,3,9,5,3,9,112,3,5,

%U 0,3,9,16,3,9,0,3,0,5,3,9,11,3,5,16,3,0,3367

%N Least m>0 such that n = y^2 - 10^x (mod m) has no solution, or 0 if no such m exists.

%C To prove that an integer n is in A051212, it is sufficient to find integers x,y such that y^2 - 10^x = n. In that case, a(n)=0. To prove that n is *not* in A051212, it is sufficient to find a modulus m for which the (finite) set of all possible values of 10^x and y^2 allows us to deduce that y^2 - 10^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

%H M. F. Hasler, <a href="/A200520/b200520.txt">Table of n, a(n) for n = 0..1000</a>

%e See A200512 for motivation and detailed examples.

%o (PARI) A200520(n,b=10,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, bx=bx*b%m), for(i=1,#qr, qr[i]<bx & next; qr[i]>bx & break; next(3))); return(m))}

%Y Cf. A051204-A051221, A200505-A200524.

%K nonn

%O 0,2

%A _M. F. Hasler_, Nov 18 2011