A200507 -id:A200507 - cp's OEIS frontend

A200505 Least m>0 such that n = 5^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 4, 4, 0, 0, 4, 4, 5, 0, 4, 4, 24, 5, 4, 4, 0, 15, 4, 4, 75, 0, 4, 4, 0, 0, 4, 4, 5, 39, 4, 4, 15, 5, 4, 4, 24, 35, 4, 4, 175, 31, 4, 4, 0, 39, 4, 4, 5, 0, 4, 4, 35, 5, 4, 4, 21, 55, 4, 4, 24, 0, 4, 4, 31, 39, 4, 4, 5, 399, 4, 4, 31, 5, 4, 4, 0, 15, 4, 4

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051216, i.e., not of the form 5^x-y^2. On the other hand, if there are integers x, y such that n=5^x-y^2, then we know that a(n)=0.

a(144) > 20000.

			See A200507.

M. F. Hasler, Table of n, a(n) for n = 0..143

Cf. A051204-A051221, A200505-A200524.

A200505(n,b=5,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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

a(2+4k)=a(3+4k)=4, a(8+20k)=a(13+20k)=5 for all k>=0.

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

Original entry on oeis.org

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))}

A200523 Least m>0 such that n = 3^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 0, 0, 8, 0, 8, 9, 0, 0, 12, 0, 8, 9, 8, 20, 9, 0, 0, 12, 8, 80, 8, 0, 45, 9, 0, 0, 8, 80, 8, 9, 0, 45, 9, 20, 8, 21, 8, 80, 9, 80, 28, 9, 8, 0, 8, 0, 91, 9, 20, 36, 8, 0, 8, 12, 0, 80, 9, 80, 8, 9, 8, 28, 15, 0, 91, 9, 8, 45, 8, 0, 0, 15, 0, 20, 8, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051214, i.e., not of the form 3^x-y^2. On the other hand, if there are integers x, y such that n=3^x-y^2, then we know that a(n)=0.

			See A200507 for developed examples.
Some of the larger values include a(107)=17732, a(146)=1924, a(347)=4400, a(416)=2044, a(458)>30000.

M. F. Hasler, Table of n, a(n) for n = 0..457

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

A200523(n,b=3,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).

A200508 Least m>0 such that n = 8^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 4, 7, 0, 7, 4, 0, 0, 7, 4, 8, 7, 20, 4, 0, 7, 7, 4, 7, 9, 16, 4, 7, 7, 16, 4, 8, 0, 9, 4, 7, 9, 7, 4, 8, 48, 7, 4, 0, 7, 9, 4, 8, 7, 7, 4, 7, 0, 20, 4, 7, 7, 12, 4, 0, 9, 16, 4, 7, 0, 7, 4, 0, 0, 7, 4, 8, 7, 16, 4, 0, 7, 7, 4, 7, 32, 9, 4, 7, 7, 44, 4

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051219, i.e., not of the form 8^x-y^2. On the other hand, if n is in A051219, i.e., there are integers x, y such that n=8^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, A200505-A200524.

A200508(n,b=8,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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

A200509 Least m>0 such that n = 9^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 4, 4, 8, 0, 4, 4, 0, 0, 4, 4, 8, 9, 4, 4, 9, 0, 4, 4, 8, 80, 4, 4, 45, 9, 4, 4, 8, 80, 4, 4, 0, 45, 4, 4, 8, 21, 4, 4, 9, 61, 4, 4, 8, 0, 4, 4, 45, 9, 4, 4, 8, 0, 4, 4, 0, 80, 4, 4, 8, 9, 4, 4, 15, 0, 4, 4, 8, 45, 4, 4, 0, 15, 4, 4, 8, 0, 4, 4, 0, 0, 4

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051220, i.e., not of the form 9^x-y^2. On the other hand, if n is in A051220, i.e., there are integers x, y such that n=9^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, A200505-A200524.

A200509(n,b=9,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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

A200510 Least m>0 such that n = 10^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 3, 5, 9, 3, 0, 9, 3, 0, 0, 3, 35, 5, 3, 11, 9, 3, 5, 0, 3, 16, 9, 3, 11, 9, 3, 20, 5, 3, 16, 9, 3, 5, 9, 3, 0, 11, 3, 0, 9, 3, 20, 5, 3, 32, 11, 3, 5, 9, 3, 0, 9, 3, 28, 37, 3, 11, 5, 3, 200, 9, 3, 5, 0, 3, 16, 9, 3, 16, 9, 3, 35, 5, 3, 0, 9, 3, 5, 9

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051221, i.e., not of the form 10^x-y^2. On the other hand, if n is in A051221, i.e., there are integers x, y such that n=10^x-y^2, then we know that a(n)=0.

Similar to the comment in A200522, it is likely (but still unproven) that a(n) = 0 if and only if n is in A051221. The extension by Azuma confirms that this holds for n = 0..2000. - Seiichi Azuma, Apr 04 2025

			See A200507.

Seiichi Azuma, Table of n, a(n) for n = 0..2000

Cf. A051204-A051221, A200505-A200524.

A200510(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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

a(111)=11111.

cp's OEIS Frontend

A200505 Least m>0 such that n = 5^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A200523 Least m>0 such that n = 3^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A200508 Least m>0 such that n = 8^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A200509 Least m>0 such that n = 9^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A200510 Least m>0 such that n = 10^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links