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

1, 2, 2, 5, 8, 3, 3, 10, 7, 18, 15, 4, 4, 17, 170, 9, 55, 197, 24, 5, 5, 26, 127, 70, 11, 1520, 17, 23, 35, 6, 6, 37, 25, 19, 32, 13, 3482, 199, 161, 24335, 48, 7, 7, 50, 649, 182, 485, 89, 15, 151, 99, 530, 31, 29718, 63, 8, 8, 65, 48842, 33, 7775, 251, 3480, 17, 1068, 43, 26, 57799, 351, 53, 80, 9, 9, 82, 55, 378, 10405, 28, 197, 500, 19, 1574, 1151, 12151, 2143295, 39, 49, 5604, 99, 10, 10, 101, 227528

Offset: 1

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• nice

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 a(n) and b(n) numbers for the +1 option are 2*a(n)^2+1 and 2*a(n)*b(n), respectively (see Perron I, pp. 94,95).

If d(n)=A000037(n)= k^2+1, k=1,2,.., then the a^2 - d(n)*b^2 = -1 Pell equation has the minimal solution a(n)=k and b(n)=1. If d(n)=A000037(n)= k^2-1, k=2,3,..., then the a^2 - d(n)*b^2 = +1 Pell equation has the minimal solution a=k and b=1.

The general integer solutions (up to signs) of Pell equation a^2 - d(n)*b^2 = +1 with d(n)=A000037(n), but not from A003654, are a(n,p)= T(p+1,a(n)) and b(n,p)= b(n)*S(p,2*a(n)), p=0,1,... If d(n)=A000037(n) is also from A003654 then these solutions are a(n,p)= T(p+1,2*a(n)^2+1) and b(n,p)= 2*a(n)*b(n)*S(p,2*(2*a(n)^2+1)), p=0,1,... Here T(n,x), resp. S(n,x) := U(n,x/2), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.

The general integer solutions (up to signs) of the Pell equation a^2 - d(n)*b^2 = -1 with d(n)=A000037(n)= A003654(k), for some k>=1, are a(n,p) = a(n)*(S(n,2*(2*a(n)^2)+1) + S(n-1,2*(2*a(n)^2)+1)) and b(n,p) = b(n)*(S(n,2*(2*a(n)^2)+1) - S(n-1,2*(2*a(n)^2)+1)) with the S(n,x) := U(n,x/2) Chebyshev polynomials. S(-1,x) := 0.

If the trivial solution x=1, y=0 is included, the sequence becomes A006702. - T. D. Noe, May 17 2007

Examples

			d=10=A000037(7)=A003654(3), therefore a(7)^2=10*b(7)^2 -1, i.e. 3^2=10*1^2 -1 and 2*a(7)^2+1=19 and 2*a(7)*b(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 a^2 - d*b^2 = -1 Pell equation and a(8)=10 and b(8)=A077233(8)=3 satisfy 10^2 - 11*3^2 = +1.
10=d(7)=A000037(7)=A003654(3)=3^2+1 hence a(7)=3 and b(7)=1 are the smallest numbers satisfying a^2-10*b^2=-1.
8=d(6)=A000037(6)=3^2-1 (not in A003654) hence a(6)=3 and b(6)=1 are the smallest numbers satisfying a^2-8*b^2=+1.
		

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]
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A000037, A003654, A003814, A006702, A033313, A077233.

Programs

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]];
a[n_] := a[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

Formula

a(n)=sqrt(A000037(n)*A077233(n)^2 + (-1)^(c(n))) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.