A200522 -id:A200522 - 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))}

A051213 Numbers of the form 2^x-y^2 >= 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 12, 15, 16, 23, 28, 31, 32, 39, 47, 48, 55, 60, 63, 64, 71, 79, 87, 92, 103, 112, 119, 124, 127, 128, 135, 151, 156, 175, 183, 188, 192, 199, 207, 220, 223, 231, 240, 247, 252, 255, 256, 271, 284, 287, 295, 316, 343, 348, 367, 368, 375, 391, 399, 412, 431, 448

Offset: 1

David W. Wilson

nonn

Is 519 in this sequence? Then this is the value of a(73), else it is 527, after which the sequence goes on with 540, 583, 604, 615, 623, 624,... - M. F. Hasler, Oct 09 2014
From R. J. Mathar, Oct 21 2014: (Start)
519 is not in the sequence. [Proof: Consider 2^x-519=y^2 and both sides modulo 3.
Then 2^x-519 = 1,2,1,2.... (mod 3) for x>=0 and y^2=0,1,1,0,1,1,... (mod 3) for y>=0.
For moduli to match (i.e, both 1), x must be even. Then 2^x is the square of the integer y=2^(x/2). (Note that this reference does not work in integers if x is odd).
The next smaller perfect square is (y-1)^2 = (2^(x/2)-1)^2 = 2^x-2^(1+x/2)+1 .
This must be >=2^x-519 to have a solution, so -2^(1+x/2)+1 >= -519
implies 2^(1+x/2)-1 <= 519, which implies 1+x/2 <= 9.02 and x<=16.
One can check numerically that the range 0<=x<=16 do not form perfect squares 2^x-519.] (End)

M. F. Hasler, Table of n, a(n) for n = 1..72
J. Cohn, The diophantine equation x^2+C=y^n, Acta Arithm. 65 (4) (1993) 367-381
Fadwa S. Abu Muriefah, Yann Bugeaud, The diophantine equation x^2+c=y^n: a brief overview, Rev. Colomb. Matem. 40 (1) (2006) 31-37

Cf. A201125.

max = 1000; Clear[f]; f[m_] := f[m] = Select[Table[2^x - y^2, {x, 0, m}, {y, 0, Ceiling[2^(x/2)]}] // Flatten // Union, 0 <= # <= max &]; f[1]; f[m = 2]; While[f[m] != f[m - 1], m++]; Print["m = ", m]; A051213 = f[m] (* Jean-François Alcover, May 13 2017 *)

is_A051213(n)=!A200522(n) \\ M. F. Hasler, Oct 09 2014

More terms from M. F. Hasler, Oct 09 2014

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).

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.

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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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A051213 Numbers of the form 2^x-y^2 >= 0.

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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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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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