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A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

%I A254936 #21 May 23 2025 01:16:50
%S A254936 9,11,13,19,25,15,21,23,35,41,25,21,37,49,23,39,29,25,57,35,27,59,65,
%T A254936 33,43,29,49,55,51,41,37,69,81,39,59,35,65,71,77,83,51,67,47,43,79,39,
%U A254936 97,69,49,59,41,87,93,61,47,57,89,53,101,79,59,85,55,91,103,81,115,53,49,63,83,73,111,59
%N A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).
%C A254936 The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007519(n) = -(1 + 8*A005123(n)) is given in A254937(n).
%C A254936 For comments and the Nagell reference see A254934.
%H A254936 M. F. Hasler, <a href="/A254936/b254936.txt">Table of n, a(n) for n = 1..1000</a>, May 22 2025
%F A254936 a(n)^2 - 2*A254937(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
%F A254936 a(n) = -(3*A254934(n) - 4*A254935(n)), n >= 1.
%e A254936 The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007519(n) appears as first entry):
%e A254936   [17, [9, 7]], [41, [11, 9]], [73, [13, 11]],
%e A254936   [89, [19, 15]], [97, [25, 19]], [113, [15, 13]],
%e A254936   [137, [21, 17]], [193, [23, 19]], [233, [35, 27]],
%e A254936   [241, [41, 31]], [257, [25, 21]], [281, [21, 19]],
%e A254936   [313, [37, 29]], [337, [49, 37]], [353, [23, 21]],
%e A254936   [401, [39, 31]], [409, [29, 25]], [433, [25, 23]],
%e A254936   [449, [57, 43]], [457, [35, 29]], [521, [27, 25]],
%e A254936   [569, [59, 45]], [577, [65, 49]], [593, [33, 29]],
%e A254936   [601, [43, 35]], [617, [29, 27]], [641, [49, 39]], ...
%e A254936 a(4) = -(3*3 - 4*7) = 28 - 9 = 19.
%o A254936 (PARI) apply( {A254936(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - _M. F. Hasler_, May 22 2025
%Y A254936 Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).
%Y A254936 Cf. A254937 (corresponding y2-values), A254934 (x1 values), A254935 (y1 values),  A255233 (same for primes == 7 mod 8), A255247.
%K A254936 nonn,easy
%O A254936 1,1
%A A254936 _Wolfdieter Lang_, Feb 18 2015
%E A254936 More terms from _M. F. Hasler_, May 22 2025