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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). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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.

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.

EXAMPLE

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.

MATHEMATICA

d[n_] := d[n] = n + Floor[Sqrt[n] + 1/2]; r[n_, a_] := Reduce[lhs = a^2 - d[n]*b^2; b > 0 && (lhs == 1 || lhs == -1) , {b}, Integers]; r[n_] := For[a = 1, True, a++, If[r[n, a] =!= False, Return[a]]]; A077232 = Table[a = r[n]; Print["a(", n, ") = ", a]; a, {n, 1, 93}] (* Jean-François Alcover, May 22 2012 *)

CROSSREFS

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

Sequence in context: A265819 A254746 A011021 * A193891 A193906 A224791

Adjacent sequences:  A077229 A077230 A077231 * A077233 A077234 A077235

KEYWORD

nonn,nice

AUTHOR

Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified August 3 04:30 EDT 2020. Contains 336197 sequences. (Running on oeis4.)