A051204 -id:A051204 - cp's OEIS frontend

A248344 Primes of the form x^2 - 2^y.

2, 3, 5, 7, 17, 23, 41, 47, 73, 79, 89, 97, 113, 137, 167, 193, 223, 233, 257, 281, 313, 353, 359, 401, 409, 433, 439, 449, 457, 521, 577, 593, 601, 617, 727, 761, 809, 839, 857, 929, 953, 977, 1009, 1087, 1097, 1193, 1201, 1217, 1223, 1361, 1367, 1433, 1489, 1553, 1609, 1697, 1721

Offset: 1

Juri-Stepan Gerasimov, Oct 05 2014

nonn

Primes in A051204.

			17 is in this sequence because 17 = 5^2 - 2^3 = 7^2 - 2^5 = 9^2 - 2^6 = 23^2 - 2^9.

Cf. A051204, A247763.

a(54) corrected by Charles R Greathouse IV, Nov 26 2014

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.

A200513 Least m>0 such that n = y^2 - 3^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 8, 0, 8, 9, 0, 0, 0, 0, 8, 9, 8, 0, 9, 0, 0, 12, 8, 0, 8, 28, 0, 9, 0, 20, 8, 0, 8, 9, 20, 80, 9, 0, 8, 0, 8, 0, 9, 63, 0, 9, 8, 80, 8, 20, 0, 9, 0, 28, 8, 63, 8, 12, 0, 0, 9, 36, 8, 9, 8, 0, 12, 0, 532, 9, 8, 80, 8, 108, 20, 15, 0, 0, 8, 63, 8, 9, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051205, it is sufficient to find integers x,y such that y^2 - 3^x = n. In that case, a(n)=0. To prove that n is *not* in A051205, it is sufficient to find a modulus m for which the (finite) set of all possible values of 3^x and y^2 (mod m) allows us to deduce that y^2 - 3^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

			See A200512.

Cf. A051204-A051221, A200505-A200520.

A200513(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))}

A200514 Least m>0 such that n = y^2 - 4^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 16, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 16, 0, 3, 4, 16, 3, 40, 4, 3, 0, 0, 3, 0, 0, 3, 4, 16, 3, 63, 4, 3, 63, 0, 3, 20, 0, 3, 4, 20, 3, 40, 4, 3, 80, 0, 3, 16, 0, 3, 4, 0, 3, 0, 4, 3, 0, 40, 3, 16, 80, 3, 4, 16, 3, 0, 4, 3, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051206, it is sufficient to find integers x,y such that y^2 - 4^x=n. In that case, a(n)=0. To prove that n is *not* in A051206, it is sufficient to find a modulus m for which the (finite) set of all possible values of 4^x and y^2 (mod m) allows us to deduce that y^2 - 4^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

			See A200512.

M. F. Hasler, Table of n, a(n) for n = 0..688

Cf. A051204-A051221, A200505-A200520.

A200514(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))}

A200515 Least m>0 such that n = y^2 - 5^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 4, 4, 0, 0, 4, 4, 5, 0, 4, 4, 0, 5, 4, 4, 0, 24, 4, 4, 0, 0, 4, 4, 35, 0, 4, 4, 5, 15, 4, 4, 0, 5, 4, 4, 0, 24, 4, 4, 0, 24, 4, 4, 15, 0, 4, 4, 5, 0, 4, 4, 0, 5, 4, 4, 39, 0, 4, 4, 0, 24, 4, 4, 0, 24, 4, 4, 5, 35, 4, 4, 0, 5, 4, 4, 0, 0, 4, 4, 63, 0, 4, 4

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051207, it is sufficient to find integers x,y such that y^2 - 5^x = n. In that case, a(n)=0. To prove that n is *not* in A051207, it is sufficient to find a modulus m for which the (finite) set of all possible values of 5^x and y^2 (mod m) allows us to deduce that y^2 - 5^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

			See A200512 for motivation and detailed examples.

Cf. A051204-A051221, A200505-A200520.

A200515(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,y,y^2-n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

A200515(A051207(k))=0 for all k in N.

A200515(1+4k)=A200515(2+4k)=4 for all k>=0.

A200516 Least m>0 such that n = y^2 - 6^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

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, 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, 5, 5, 9, 36, 16, 5, 5, 0, 7, 0, 5, 5, 32, 24, 0, 5, 5

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

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.

			See A200512 for motivation and detailed examples.

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

Cf. A051204-A051221, A200505-A200520.

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

A200517 Least m>0 such that n = y^2 - 7^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

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, 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, 8, 0, 3, 16, 28, 3, 7, 16, 3, 16, 0, 3, 0, 7, 3, 19, 8, 3

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

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.

			See A200512 for motivation and detailed examples.

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

Cf. A051204-A051221, A200505-A200520.

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

A200518 Least m>0 such that n = y^2-8^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 4, 0, 7, 7, 4, 9, 0, 7, 4, 7, 7, 8, 4, 0, 7, 0, 4, 7, 32, 8, 4, 7, 0, 7, 4, 16, 0, 8, 4, 9, 7, 7, 4, 0, 0, 7, 4, 7, 7, 0, 4, 9, 7, 8, 4, 7, 0, 9, 4, 7, 9, 7, 4, 12, 0, 0, 4, 16, 7, 7, 4, 0, 0, 7, 4, 7, 7, 8, 4, 16, 7, 0, 4, 7, 9, 8, 4, 7, 0, 7, 4, 32

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If a(n)>0, this proves that n cannot be a member of A051210, i.e., cannot be written as y^2-8^x. To prove that an integer n is in A051210, it is sufficient to find integers x,y such that y^2-8^x=n. In that case, a(n)=0.

			See A200512 for motivation and detailed examples.

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

Cf. A051204-A051221, A200505-A200520.

A200518(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,y,y^2-n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

cp's OEIS Frontend

A248344 Primes of the form x^2 - 2^y.

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

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

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

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A200513 Least m>0 such that n = y^2 - 3^x (mod m) has no solution, or 0 if no such m exists.

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A200514 Least m>0 such that n = y^2 - 4^x (mod m) has no solution, or 0 if no such m exists.

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A200515 Least m>0 such that n = y^2 - 5^x (mod m) has no solution, or 0 if no such m exists.

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A200516 Least m>0 such that n = y^2 - 6^x (mod m) has no solution, or 0 if no such m exists.

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A200517 Least m>0 such that n = y^2 - 7^x (mod m) has no solution, or 0 if no such m exists.