cp's OEIS Frontend

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

%I A200522 #28 Apr 22 2025 02:48:53
%S A200522 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,
%T A200522 16,40,0,20,12,0,0,52,20,15,80,16,16,0,112,36,12,56,33,28,28,0,0,20,
%U A200522 15,40,128,16,12,0,117,48,16,24,0,44,28,0,0,15,12,40,63
%N A200522 Least m>0 such that n = 2^x-y^2 (mod m) has no solution, or 0 if no such m exists.
%C A200522 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.
%C A200522 a(519) > 20000 if it is nonzero.
%C A200522 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
%C A200522 a(519) = 131235. - _Seiichi Azuma_, Apr 05 2025
%H A200522 Seiichi Azuma, <a href="/A200522/b200522.txt">Table of n, a(n) for n = 0..1000</a> (terms up to a(518) from M. F. Hasler)
%e A200522 See A200507 for motivation and examples.
%o A200522 (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, bx=bx*b%m), for(i=1,#qr, qr[i]<bx & next; qr[i]>bx & break; next(3))); return(m))}
%Y A200522 Cf. A051204-A051221, A200523, A200524, A200505-A200520.
%K A200522 nonn
%O A200522 0,6
%A A200522 _M. F. Hasler_, Nov 18 2011