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%I #10 Jan 03 2016 04:36:45
%S 0,5,5,0,7,9,5,5,0,0,0,5,5,0,9,0,5,5,7,0,9,5,5,9,0,7,5,5,0,9,0,5,5,36,
%T 16,0,5,5,9,7,0,5,5,0,21,0,5,5,0,43,9,5,5,7,16,16,5,5,0,9,7,5,5,0,0,9,
%U 5,5,9,36,16,5,5,0,7,0,5,5,32,24,0,5,5
%N Least m>0 such that n = y^2 - 6^x (mod m) has no solution, or 0 if no such m exists.
%C To prove that an integer n is in A051208, it is sufficient to find integers x,y such that y^2 - 6^x = n. In that case, a(n)=0. To prove that n is *not* in A051208, it is sufficient to find a modulus m for which the (finite) set of all possible values of 6^x and y^2 (mod m) allows us to deduce that y^2 - 6^x can never equal n. The present sequence lists the smallest such m>0, if it exists.
%H M. F. Hasler, <a href="/A200516/b200516.txt">Table of n, a(n) for n = 0..1000</a>
%e See A200512 for motivation and detailed examples.
%o (PARI) A200516(n,b=6,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
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