A254938 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

1, 3, 1, 5, 1, 7, 5, 1, 7, 11, 3, 1, 13, 7, 5, 11, 9, 17, 5, 3, 9, 19, 7, 13, 5, 3, 7, 19, 13, 1, 9, 25, 15, 7, 23, 27, 17, 9, 21, 7, 1, 13, 19, 11, 23, 17, 31, 7, 1, 33, 11, 17, 7, 27, 5, 35, 13, 25, 19, 11, 29, 9, 17, 5, 3, 1, 27, 21, 35, 17, 23, 15, 37, 41, 21, 13, 19, 7, 3, 23, 15, 33, 13

Offset: 1

Author

Wolfdieter Lang, Feb 18 2015

Keywords

• nonn

Comments

For the corresponding term y1(n) see 2*A255232(n).

For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A255233(n) and A255234(n).

The present solutions of this first class are the smallest positive ones.

See the Nagell reference Theorem 111, p. 210, for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).

See the Nagell reference Theorem 110, p. 208, for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable and each prime from A007522 does not divide 4.

The present fundamental solutions are found according to the Nagell reference Theorem 108a, p. 206-207, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) (because odd y is out in this Pell equation). The interval to be scanned for X1(n) is [0, floor((sqrt(p(n))-1)/2)] and for Y1(n) it is [0, floor(sqrt(p(n))/2)], with p(n) = A007522(n).

The general positive proper solutions are for both classes obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental positive column vectors (x(n),y(n))^T. The n-th power is M^n = S(n-1, 6)*M - S(n-2, 6) 1_2, where 1_2 is the 2 X 2 identity matrix and S(n, 6), with S(-2, 6) = -1 and S(-1, 6) = 0, is the Chebyshev S-polynomial evaluated at x = 6, given in A001109(n).

The least positive x solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255235.

Examples

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A007522(n) is listed as first entry):
  [7, [1, 2]], [23, [3, 4]], [31, [1, 4]],
  [47, [5, 6]], [71, [1, 6]], [79, [7, 8]],
  [103, [5, 8]], [127, [1, 8]], [151, [7, 10]],
  [167, [11, 12]], [191, [3, 10]], [199, [1, 10]], [223, [13, 14]], [239, [7, 12]], [263, [5, 12]], [271, [11, 14]], [311, [9, 14]], [359, [17, 18]], [367, [5, 14]], [383, [3, 14]], [431, [9, 16]], [439, [19, 20]], [463, [7, 16]], [479, [13, 18]], [487, [5, 16]], [503, [3, 16]], ...
n=1: 1^2 - 2*(2*1)^2 = 1 - 8 = -7 = -A007522(1), ...
		

References

• T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Crossrefs

Cf. A007522 (primes == 7 mod 8).

Cf. A255232 (corresponding y2 values, halved), A255233 (x2 values), A255234 (y2 values), A255235.

Cf. A254934 - A254937 (similar for primes == 1 mod 8).

Programs

PARI

apply( {A254938(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][1]}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

Formula

a(n)^2 - 2*(2*A255232(n))^2 = -A007522(n), n >= 1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

Extensions

More terms from Colin Barker, Feb 23 2015