A254760 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007519(n), n>=1 (primes congruent to 1 mod 8).

5, 7, 9, 11, 13, 11, 13, 15, 19, 21, 17, 17, 21, 25, 19, 23, 21, 21, 29, 23, 23, 31, 33, 25, 27, 25, 29, 31, 31, 29, 29, 37, 41, 31, 35, 31, 37, 39, 41, 43, 35, 39, 35, 35, 43, 35, 49, 41, 37

Offset: 1

Author

Wolfdieter Lang, Feb 10 2015

Keywords

• nonn
• easy

Comments

For the corresponding term y1(n) see 2*A254761(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 A254762(n) and 2*A254763(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 the primes from A007519 do not divide 4.

The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, 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). The intervals to be scanned are ceiling((sqrt(8 + p(n))-1)/2) <= X1(n) <= floor((sqrt(2*p(n))-1)/2), with p(n) = A007519(n), and

1 <= Y1(n) <= floor(sqrt(A005123(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 positive fundamental column vectors (x(n),y(n))^T. The n-th power 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 the ones of the first class) for the primes +1 and -1 (mod 8) together (including in the first class also the prime 2) are given in A002334. - Wolfdieter Lang, Feb 12 2015

Examples

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007519(n) as first entry):
[17, [5, 2]], [41, [7, 2]], [73, [9, 2]], [89, [11, 4]], [97, [13, 6]], [113, [11, 2]], [137, [13, 4]], [193, [15, 4]], [233, [19, 8]], [241, [21, 10]], [257, [17, 4]], [281, [17, 2]], [313, [21, 8]], [337, [25, 12]], [353, [19, 2]], [401, [23, 8]], [409, [21, 4]], ...
n=1: 5^2 - 2*2^2 = 25 - 8 = 17, ...
		

References

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

Crossrefs

Cf. A007519, A005123, 2*A254761, A254762, 2*A254763, A002334.

Formula

a(n)^2 - 2*(2*A254760(n))^2 = A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.