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A254934 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).

%I A254934 #24 May 23 2025 01:17:03
%S A254934 1,3,5,3,1,7,5,7,3,1,9,13,5,1,15,7,13,17,1,11,19,3,1,17,11,21,9,7,11,
%T A254934 17,21,5,1,23,11,27,9,7,5,3,19,11,23,27,7,31,1,13,25,19,33,7,5,21,31,
%U A254934 25,9,29,5,15,27,13,31,11,7,17,3,37,41,31,19,25,7,35,5,17,33,13,21,19,45,25,3
%N A254934 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).
%C A254934 For the corresponding term y1(n) see A254935(n).
%C A254934 For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254936(n) and A254937(n).
%C A254934 The present solutions of this first class are the smallest positive ones.
%C A254934 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).
%C A254934 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 A007519 does not divide 4.
%C A254934 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) + 1 (because even y is out in this Pell equation). The intervals to be scanned are identical for X1(n) and Y1(n), namely [0, floor((sqrt(p(n) - 1)/2)] with p(n) = A007519(n).
%C A254934 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 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).
%C A254934 The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255235.
%D A254934 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
%H A254934 M. F. Hasler, <a href="/A254934/b254934.txt">Table of n, a(n) for n = 1..1000</a>
%F A254934 a(n)^2 - 2*A254935(n)^2 = -A007519(n), n >=1, gives the smallest positive (proper) solution of this (generalized) Pell equation.
%e A254934 The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A007519(n) is listed as first entry):
%e A254934 [17, [1, 3]], [41, [3, 5]], [73, [5, 7]],
%e A254934 [89, [3, 7]], [97, [1, 7]], [113, [7, 9]],
%e A254934 [137, [5, 9]], [193, [7, 11]], [233, [3, 11]],
%e A254934 [241, [1, 11]], [257, [9, 13]], [281, [13, 15]],
%e A254934 [313, [5, 13]], [337, [1, 13]], [353, [15, 17]],
%e A254934 [401, [7, 15]], [409, [13, 17]], [433, [17, 19]],
%e A254934 [449, [1, 15]], [457, [11, 17]], [521, [19, 21]],
%e A254934 [569, [3, 17]], [577, [1, 17]], [593, [17, 21]],
%e A254934 [601, [11, 19]], [617, [21, 23]], [641, [9, 19]],
%e A254934 [673, [7, 19]], [761, [11, 21]], [769, [17, 23]],
%e A254934 ...
%e A254934 n=1: 1^2 - 2*3^2 = 1 - 18 = -17, ...
%o A254934 (PARI) apply( {A254934(n, p=A007519(n))=Set(abs(qfbsolve(Qfb(-1,0,2), p,1)))[1][1]}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - _M. F. Hasler_, May 22 2025
%Y A254934 Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).
%Y A254934 Cf. A254935 (corresponding y1 values), A254936 (x2 values), A254937 (y2 values), A254938, A255232, A255235, A254760.
%K A254934 nonn,easy
%O A254934 1,2
%A A254934 _Wolfdieter Lang_, Feb 18 2015
%E A254934 More terms from _M. F. Hasler_, May 22 2025