|
%I #10 Jan 03 2016 04:36:57
%S 0,3,0,0,3,7,8,3,0,0,3,16,7,3,8,0,3,16,0,3,8,16,3,16,0,3,7,16,3,0,8,3,
%T 0,7,3,0,8,3,8,16,3,28,0,3,8,19,3,7,0,3,20,0,3,16,7,3,100,0,3,16,8,3,
%U 8,0,3,16,28,3,7,16,3,16,0,3,0,7,3,19,8,3
%N Least m>0 such that n = y^2 - 7^x (mod m) has no solution, or 0 if no such m exists.
%C To prove that an integer n is in A051209, it is sufficient to find integers x,y such that y^2 - 7^x = n. In that case, a(n)=0. To prove that n is *not* in A051209, it is sufficient to find a modulus m for which the (finite) set of all possible values of 7^x and y^2 allows us to deduce that y^2-7^x can never equal n. The present sequence lists the smallest such m>0, if it exists.
%H M. F. Hasler, <a href="/A200517/b200517.txt">Table of n, a(n) for n = 0..1000</a>
%e See A200512 for motivation and detailed examples.
%o (PARI) A200517(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,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-A200520.
%K nonn
%O 0,2
%A _M. F. Hasler_, Nov 18 2011
|
