A200505 -id:A200505 - cp's OEIS frontend

A200520 Least m>0 such that n = y^2 - 10^x (mod m) has no solution, or 0 if no such m exists.

0, 3, 5, 0, 3, 9, 0, 3, 0, 11, 3, 9, 5, 3, 9, 0, 3, 5, 11, 3, 9, 0, 3, 9, 0, 3, 0, 5, 3, 9, 16, 3, 5, 20, 3, 0, 1001, 3, 9, 0, 3, 9, 5, 3, 0, 56, 3, 5, 0, 3, 9, 11, 3, 11, 0, 3, 9, 5, 3, 9, 112, 3, 5, 0, 3, 9, 16, 3, 9, 0, 3, 0, 5, 3, 9, 11, 3, 5, 16, 3, 0, 3367

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051212, it is sufficient to find integers x,y such that y^2 - 10^x = n. In that case, a(n)=0. To prove that n is *not* in A051212, it is sufficient to find a modulus m for which the (finite) set of all possible values of 10^x and y^2 allows us to deduce that y^2 - 10^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-A200524.

A200520(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,y,y^2-n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

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))}

A200507 Least m>0 such that n = 7^x-y^2 (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 3, 0, 8, 3, 0, 0, 3, 7, 8, 3, 8, 0, 3, 16, 7, 3, 8, 0, 3, 16, 28, 3, 0, 16, 3, 16, 8, 3, 7, 16, 3, 0, 8, 3, 8, 7, 3, 28, 0, 3, 8, 16, 3, 0, 19, 3, 0, 0, 3, 7, 8, 3, 0, 20, 3, 16, 7, 3, 8, 100, 3, 16, 35, 3, 8, 28, 3, 16, 20, 3, 7, 16, 3, 16, 8, 3, 28

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051218, i.e., not of the form 7^x-y^2. On the other hand, if there are integers x, y such that n=7^x-y^2, then we know that a(n)=0.

a(432) is at least of order 10^5.

			a(2)=3 since quadratic residues mod 3 (i.e. possible values for y^2 mod 3) are {0,1}, and 7^x is always congruent to 1 (mod 3), therefore there cannot be any (x,y) such that 7^x-y^2 = 2. The modulus m=3 is the least number for which this equation has no solution in Z/mZ: For m=1 the equation is always true, and for m=2 one always has the solution x=0 and y=0 (for even n) or y=1 (for odd n).

M. F. Hasler, Table of n, a(n) for n = 0..431

Cf. A051204-A051221, A200505-A200524.

A200507(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,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

A200512 Least m>0 such that n = y^2 - 2^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 15, 16, 0, 24, 0, 0, 0, 0, 12, 24, 0, 0, 16, 0, 0, 15, 16, 16, 0, 40, 12, 20, 0, 0, 0, 0, 0, 24, 16, 28, 15, 0, 12, 16, 64, 0, 20, 0, 0, 0, 20, 20, 39, 40, 12, 15, 0, 0, 16, 16, 0, 24, 0, 0, 0, 0, 12, 24, 0, 40, 15, 20, 112, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051204, it is sufficient to find (x,y) such that y^2 - 2^x = n. In that case, a(n)=0. To prove that n is *not* in A051204, it is sufficient to find a modulus m for which the (finite) set of all possible values of 2^x and y^2 allows us to deduce that y^2 - 2^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

			a(0)=a(1)=0 because 0=1^2-2^0 and 1=3^2-2^3 are in A051204. Similarly, n=2 through n=5 are in A051204, i.e., there are (x,y) such that n=y^2-2^x, but for n=6 such (x,y) cannot exist:
a(6)=12 because for all m<12 the equation y^2-2^x = 6 has a solution (mod m), but not so for m=12: Indeed, y^2 equals 0, 1, 4 or 9 (mod 12) and 2^x equals 1, 2, 4 or 8 (mod 12). Therefore y^2-2^x can never be equal to 6, else the equality would also hold modulo m=12.

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

Cf. A051204-A051221, A200505-A200520.

A200512(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))}

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

A200520 Least m>0 such that n = y^2 - 10^x (mod m) has no solution, or 0 if no such m exists.

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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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A200507 Least m>0 such that n = 7^x-y^2 (mod m) has no solution, or 0 if no such m exists.

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A200512 Least m>0 such that n = y^2 - 2^x (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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A200523 Least m>0 such that n = 3^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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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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