A048941 -id:A048941 - cp's OEIS frontend

A048942 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

2, 2, 1, 4, 6, 1, 2, 6, 1, 1, 8, 2, 2, 8, 78, 1, 1, 84, 10, 2, 2, 10, 3, 1, 4, 546, 1, 8, 12, 2, 2, 12, 8, 1, 10, 4, 1062, 3, 1, 7176, 14, 2, 2, 14, 5, 1, 132, 24, 4, 40, 26, 138, 1, 5, 16, 2, 2, 16, 11934, 1, 3, 60, 826, 4, 250, 10, 6, 39, 1, 12, 18, 2, 2, 18

Offset: 1

Eric W. Weisstein

nonn

From Sean A. Irvine, Jul 16 2021: (Start)
These values are computed by Algorithm 5.7.2 in Cohen.
Other methods of computation (see A346420) give different results, with the first difference at n=14.
(End)
a(n) is the smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = +-4, where D = A000037(n). - Jinyuan Wang, Sep 08 2021

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993.

Jinyuan Wang, 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]
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A007913, A048941, A346419, A346420.

a(n) = my(A, D=n+(1+sqrtint(4*n))\2, d=sqrtint(D), p, q, t, u1, u2, v1, v2); if(d%2==D%2, p=d, p=d-1); u1=-p; u2=2; v1=1; v2=0; q=2; while(v2==0 || q!=t, A=(p+d)\q; t=p; p=A*q-p; if(t==p && v2!=0, return(2*u2*v2/q), t=A*u2+u1; u1=u2; u2=t; t=A*v2+v1; v1=v2; v2=t; t=q; q=(D-p^2)/q)); (u1*v2+u2*v1)/q; \\ Jinyuan Wang, Sep 08 2021

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 16 2021

A346419 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Original entry on oeis.org

2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 2, 340, 1, 5, 394, 48, 10, 10, 4, 16, 5, 22, 3040, 2, 46, 70, 12, 12, 74, 50, 6, 64, 26, 6964, 20, 1, 48670, 96, 4, 2, 100, 3, 7, 10, 178, 30, 302, 198, 1060, 8, 39, 126, 16, 16, 130, 97684, 8, 25, 502, 6960, 2, 2136, 86, 4, 340, 9, 106, 160, 1, 18, 164, 5, 9, 20810

Offset: 1

Sean A. Irvine

nonn

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A346420, A048941, A048942.

nonSquares = Select[Range[72], !IntegerQ[Sqrt[#]]& ]; 2*NumberFieldFundamentalUnits[Sqrt[#]][[1, 2, 1]] & /@ nonSquares (* Jean-François Alcover, Nov 09 2012 *)

for(n=1,30,if(!issquare(n),print1(abs(2*polcoeff(lift(bnfinit(x^2-n).fu[1]),0)),","))) /* Ralf Stephan, Sep 18 2013; updated by Michel Marcus, Jun 25 2020 */

A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Original entry on oeis.org

2, 2, 1, 4, 6, 2, 2, 6, 2, 1, 8, 2, 2, 2, 78, 1, 1, 84, 10, 4, 2, 2, 6, 1, 4, 546, 2, 8, 12, 2, 2, 12, 8, 2, 10, 4, 1062, 6, 1, 7176, 14, 2, 2, 14, 1, 1, 4, 24, 8, 40, 26, 138, 2, 5, 16, 6, 2, 16, 11934, 2, 3, 60, 826, 2, 250, 10, 2, 78, 1, 12, 18, 1, 2, 18, 1, 1, 2244, 6, 84

Offset: 1

Sean A. Irvine

nonn

The radical part is actually sqrt(A007913(A000037(n))) where A007913(m) is the squarefree part of m. - Michel Marcus, Jun 26 2020

How does this sequence differ from A048942? The definitions of both sequences are identical, but the second comment in A048942 states the terms differ from n = 14 onwards. - Felix Fröhlich, Jun 16 2022

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental Unit.

Cf. A000037, A007913, A346419, A048941, A048942.

f(n) = {if (issquare(n), return (0)); if (!issquarefree(n), m = core(n), m = n); my(u = abs(2*polcoeff(lift(bnfinit(x^2-m, 1).fu[1]), 0))); if (u^2==1, return (1)); if (u^2==4, return (sqrtint((u^2+4)/m));); if (u^2 < 4, return((u^2+4)/n)); my(v2 = [(u^2-4)/m, (u^2+4)/m]); sqrtint(vecmin(select(x->denominator(x)==1, v2)));}
lista(nn) = apply(f, select(x->!issquare(x), [1..nn])); \\ Michel Marcus, Jun 25 2020; corrected Jun 16 2022

A107999 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Original entry on oeis.org

37, 101, 141, 189, 197, 269, 325, 333, 349, 373, 381, 389, 405, 485, 557, 573, 677, 701, 709, 757, 781, 813, 829, 877, 885, 901, 909, 925, 933, 973, 997, 1053, 1149, 1157, 1173, 1213, 1269, 1293, 1301, 1325, 1389, 1405, 1421, 1445, 1485, 1605, 1613, 1701, 1717

Offset: 1

Steven Finch, Jun 13 2005

nonn

C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966, section 256 VI, pp. 276-277.

A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod. 8), J. Reine Angew. Math. 53 (1857) 369-371.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
N. Ishii, P. Kaplan and K. S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.

Cf. A048941, A048942, A107996, A108160.

More terms from Jinyuan Wang, Sep 08 2021

A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Original entry on oeis.org

37, 101, 141, 197, 269, 349, 373, 381, 389, 485, 557, 573, 677, 701, 709, 757, 781, 813, 829, 877, 885, 901, 933, 973, 997, 1149, 1157, 1173, 1213, 1293, 1301, 1389, 1405, 1605, 1613, 1717, 1757, 1765, 1861, 1885, 1893, 1901, 1909, 1949, 1973, 2069, 2077, 2093

Offset: 1

Steven Finch, Jun 13 2005

nonn

C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966, section 256 VI, pp. 276-277.

Florian Breuer and James Punch, Quadratic units and cubic fields, arXiv:2507.06579 [math.NT], 2025. See p. 1.
Arthur Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Noburo Ishii, Pierre Kaplan and Kenneth S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.

Cf. A048941, A048942, A107997, A107999.

More terms from Jinyuan Wang, Sep 08 2021

cp's OEIS Frontend

A048942 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

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A346419 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

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A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A107999 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

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Author

Keywords

References

Links

Crossrefs

Extensions