A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

1, 1, 1, 2, 3, 1, 1, 3, 2, 5, 4, 1, 1, 4, 39, 2, 12, 42, 5, 1, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9, 1, 1, 9, 6, 41, 1122, 3, 21, 53, 2, 165, 120, 1260, 221064, 4, 5, 569, 10, 1, 1, 10, 22419

Offset

1,4

Comments

If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal b(n) and a(n) numbers for the +1 option are 2*b(n)^2 + 1 and 2*b(n)*a(n), respectively (see Perron I, pp. 94,p5).

For general integer solutions see A077232 comments.

If the trivial solution x=1, y=0 is included, the sequence becomes A006703. - T. D. Noe, May 17 2007

References

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [Paul Curtz, Apr 10 2019]

Formula

a(n)=sqrt((A077232(n)^2 - (-1)^(c(n)))/A000037(n)) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.

Example

d=10=A000037(7)=A003654(3), therefore a(7)=1 and b(7)=A077232(7)=3 give 3^2=10*1^2 -1 and 2*b(7)^2+1=19 and 2*b(7)*a(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the b^2 - d*a^2 = -1 Pell equation and a(8)=3 and b(8)=A077232(8)=10 satisfy 10^2 - 11*3^2 = +1. See A077232 for further examples.

Mathematica

nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
nonSquare[n_] := n + Round[Sqrt[n]];
b[n_] := b[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
a[n_] := If[n == 1, 1, SelectFirst[{Sqrt[(b[n]^2 - 1)/nonSquare[n]], Sqrt[(b[n]^2 + 1)/nonSquare[n]]}, IntegerQ]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

Crossrefs

Cf. A000037, A003654, A003814, A033317, A077232.

Sequence in context: A296659 A270823 A067627 * A282290 A178795 A123185

Adjacent sequences: A077230 A077231 A077232 * A077234 A077235 A077236

Keyword

nonn,nice

Author

Wolfdieter Lang, Nov 08 2002

Status

approved