A200523 -id:A200523 - cp's OEIS frontend

A200524 Least m>0 such that n = 4^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 3, 0, 0, 3, 4, 0, 3, 16, 4, 3, 0, 20, 3, 0, 0, 3, 4, 56, 3, 16, 4, 3, 80, 16, 3, 40, 0, 3, 4, 0, 3, 20, 4, 3, 64, 16, 3, 0, 63, 3, 4, 56, 3, 28, 4, 3, 0, 20, 3, 40, 63, 3, 4, 0, 3, 16, 4, 3, 0, 28, 3, 0, 0, 3, 4, 40, 3, 16, 4, 3, 85, 16, 3, 56, 63, 3

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051215, i.e., not of the form 4^x-y^2. On the other hand, if there are integers x, y such that n=4^x-y^2, then we know that a(n)=0.

Some of the larger values include a(303)= 1387, a(423)=1687, a(447)=2047, a(519)>30000.

			See A200507 for motivation and examples.

M. F. Hasler, Table of n, a(n) for n = 0..518

Cf. A051204-A051221, A200522, A200523, A200505-A200520.

A200524(n,b=4,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 & break; next(3))); return(m))}

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

Original entry on oeis.org

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, 16, 40, 0, 20, 12, 0, 0, 52, 20, 15, 80, 16, 16, 0, 112, 36, 12, 56, 33, 28, 28, 0, 0, 20, 15, 40, 128, 16, 12, 0, 117, 48, 16, 24, 0, 44, 28, 0, 0, 15, 12, 40, 63

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

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.
a(519) > 20000 if it is nonzero.
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
a(519) = 131235. - Seiichi Azuma, Apr 05 2025

			See A200507 for motivation and examples.

Seiichi Azuma, Table of n, a(n) for n = 0..1000 (terms up to a(518) from M. F. Hasler)

Cf. A051204-A051221, A200523, A200524, A200505-A200520.

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 & break; next(3))); return(m))}

A200506 Least m>0 such that n = 6^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 0, 5, 5, 0, 0, 9, 5, 5, 7, 0, 63, 5, 5, 36, 9, 7, 5, 5, 0, 44, 9, 5, 5, 9, 16, 0, 5, 5, 16, 7, 0, 5, 5, 0, 0, 21, 5, 5, 9, 16, 16, 5, 5, 7, 12, 0, 5, 5, 28, 36, 7, 5, 5, 12, 192, 16, 5, 5, 37, 9, 16, 5, 5, 24, 7, 9, 5, 5, 9, 0, 0, 5, 5, 36, 9, 52, 5, 5

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051217, i.e., not of the form 6^x-y^2. On the other hand, if there are integers x, y such that n=6^x-y^2, then we know that a(n)=0.

			See A200507.

M. F. Hasler, Table of n, a(n) for n = 0..1000

Cf. A051204-A051221, A200522, A200523, A200524, A200505-A200520.

A200506(n,b=6,p=3)={ my( x=0, qr, bx, seen ); for( m=2,9e9, 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))}

a(3+5k)=a(4+5k)=5, a(10+35k)=a(17+35k)=a(31+35k)=7 for all k>=0.

a(n)=9 for n=7, 16, 22, 70, 76 and 85 (mod 90).

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A200524 Least m>0 such that n = 4^x-y^2 (mod m) has no solution, or 0 if no such m exists.

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

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A200506 Least m>0 such that n = 6^x-y^2 (mod m) has no solution, or 0 if no such m exists.

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