A077058 -id:A077058 - cp's OEIS frontend

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

1, 2, 5, 3, 3, 27, 7, 37, 4, 4, 171, 22, 9, 14, 1193, 5, 5, 553, 16, 6173, 11, 45, 143, 849, 6, 6, 18339, 94, 1893, 103, 13, 33, 2353, 115, 12703, 7, 7, 67115, 701, 73, 59, 1891117, 15, 551427, 23, 49771, 39, 4105015, 8, 8, 24673, 41, 75585293, 25, 9095891, 989, 17, 386, 6445, 87, 771, 1385

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

This equation can also be written as (2*a(n) - b(n))^2 - D(n)*b(n)^2 = +4 or -4 with D(n) := A077425(n) = 1 + 4*G(n).
This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Wolfdieter Lang, Pairs of smallest positive solutions for +1 and -1 if they exist.

Cf. A077427, A217470.

g[n_] := Ceiling[Sqrt[n]] + n - 1; r[n_] := Reduce[an > 0 && bn > 0 && (an ^2 - an*bn - g[n]*bn^2 == 1 || an^2 - an*bn - g[n]*bn^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}], 1], an | bn]; a[n_] := a[n] = Min[ab[n][[All, 1]]]; Table[Print[{n, a[n]}]; a[n], {n, 1, 62}] (* Jean-François Alcover, Oct 04 2012 *)

a(n) = (A078361(n) + A077058(n)) / 2. [Max Alekseyev, Feb 06 2010]

More terms from Max Alekseyev, Feb 06 2010

a(9), a(33), a(54) corrected (after notice by Jean-François Alcover); a(58) through a(62) added. - Wolfdieter Lang, Oct 04 2012

A078361 Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).

Original entry on oeis.org

1, 3, 8, 5, 5, 46, 12, 64, 7, 7, 302, 39, 16, 25, 2136, 9, 9, 1000, 29, 11208, 20, 82, 261, 1552, 11, 11, 33710, 173, 3488, 190, 24, 61, 4354, 213, 23550, 13, 13, 124846, 1305, 136, 110, 3528264, 28, 1030190, 43, 93102, 73, 7688126, 15, 15, 46312, 77

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

Computed from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values for the Diophantine eq. x^2 - x*y - ((D(n)-1)/4)*y^2= +1, resp., -1 if D(n)=A077425(n), resp, D(n)=A077425(n) and D(n) also in A077426 (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).
The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions is a(n)=2*x(n)-y(n) and b(n)=y(n). If D(n)=A077425(n) is not in A077426 then the equation with -4 has no solution and a(n) and b(n) are the minimal solutions of the a(n)^2 - D(n)*b(n)^2 = +4 equation. If D(n)=A077425(n) is in A077426 then the a(n) and b(n) values are the minimal solution of the a(n)^2 - D(n)*b(n)^2 = -4 equation. In this case a(+,n)= a(n)^2+2 and b(+,n)=a(n)*b(n) are the minimal solution of a^2 - D(n)*b^2 = +4.
For Pell equation a^2 - D*b^2 = +4, see A077428 and A078355. For Pell equation a^2 - D*b^2 = -4, see A078356 and A078357.

			29=D(5)=A077425(5) is A077426(4), hence a(5)=5 and b(5)=A077058(5)=1 solve a^2 - 29*b^2=-4 minimally and a(+,5)=a(5)^2+2=27 with b(+,5)=a(5)*b(5)=5*1=5 solve a^2 - 29*b^2=+4 minimally. See also A077428 with companion A078355.
21=D(4)=A077425(4) is not in A077426, hence a(4)=5 and b(4)=A077058(4)=1 give the solution with minimal positive b of a^2 - 21*b^2=+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

More terms from Matthew Conroy, Apr 20 2003

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 2968, 15, 298, 16, 2, 5, 17, 1856, 1, 1, 9384, 97, 10, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 1084152, 117, 2, 746, 10, 88, 157, 126890, 1, 1, 1311, 56, 287

Offset: 1

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039955.

Subsequence of A077058 excluding terms for which A077425(n) is not squarefree. - Max Alekseyev, Dec 12 2012

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Emmanuel Vantieghem, Table of n, a(n) for n=1..1000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A053370-A053375.

2*NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 2]] & /@ Select[ Range[5, 309, 4], SquareFreeQ ]  (* Jean-François Alcover, Jul 09 2013 *)

forstep(n=5,1000,4, if(!issquarefree(n),next); print1( 2*polcoeff(lift(bnfinit(x^2-n).fu[1]),1), ", " )) /* Max Alekseyev */

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

Original entry on oeis.org

5, 8, 11, 14, 17, 19, 23, 26, 29, 32, 33, 35, 38, 40, 41, 44, 47, 50, 51, 52, 53, 54, 55, 59, 61, 62, 63, 65, 68, 71, 74, 75, 76, 77, 80, 82, 83, 85, 86, 89, 92, 94, 95, 96, 98, 101

Offset: 1

Wolfdieter Lang, Oct 04 2012

nonn

See the Perron reference for the theorem which by negation implies that this quadratic Diophantine equation has no solution if and only if A077427 is even.
See the pairs (x, y) = (A077057, A077058) which for these a(n) values are the smallest positive solutions of the Diophantine equation x^2 - x*y - a(n)*y^ = +1.
In the table on p. 108 of the Perron reference these a(n) values, called there also G, are the ones were in the third column numbers in brackets appear.
The case D = 4*G + 1 = m^2 > 1 has trivially no solutions: the equation is then X^2 - Y^2 = -4, with  X = |2*x-y|, Y = |m*y|. X and Y are either both even or both odd. In the first case one is led to the equation v^2 - w^2 = (v-w)*(v+w)= -1, with X = 2*v and Y = 2*w, and there is only the solution (v,w) = (0,1), hence 2*x = y, m*y = 2. But then m=2 and y=1 with non-integer x solution. In the other case X = 2*v+1 and Y = 2*w+1, v not w, leading to v + w + 1 = -1 with no positive integer solution. Thanks to T. D. Noe for pointing out that one has to mention that these values G = A002378(k), k>=1, with D a perfect (odd) square, are here not included.

			a(1) = 5 because 5 = A078358(4) and A077427(4) = 2, which is even.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Cf. A077057, A077058, A078358, A077427.

a(n) gives the increasingly ordered values for G from A078358 which appear at position k where A077427(k) is even, for k>=1. The next even number in A077427 appears for k = 6 and

A078358(6) = 8, hence a(2) = 8.

cp's OEIS Frontend

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A078361 Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).

Views

Author

Keywords

Comments

Examples

References

Extensions

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A217470 The Diophantine equation x^2 - x*y - G*y^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).