cp's OEIS Frontend

A031396 Numbers k such that Pell equation x^2 - k*y^2 = -1 is soluble.

%I A031396 #88 Jun 27 2025 04:28:55
%S A031396 1,2,5,10,13,17,26,29,37,41,50,53,58,61,65,73,74,82,85,89,97,101,106,
%T A031396 109,113,122,125,130,137,145,149,157,170,173,181,185,193,197,202,218,
%U A031396 226,229,233,241,250,257,265,269,274,277,281,290,293,298
%N A031396 Numbers k such that Pell equation x^2 - k*y^2 = -1 is soluble.
%C A031396 Terms are divisible neither by 4 nor by a prime of the form 4*k + 3 (although these conditions are not sufficient - compare A031398). - _Lekraj Beedassy_, Aug 17 2005
%C A031396 This is the set of integer solutions of all quadratic forms m^2*x^2 -/+ b*x + c with discriminant b^2 - 4*m^2*c = -4 where m is any term of A004613. - _Klaus Purath_, Jun 18 2025
%D A031396 Harvey Cohn, "Advanced Number Theory".
%H A031396 Ray Chandler, <a href="/A031396/b031396.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A031396 Dmitry Berdinsky and Prohrak Kruengthomya, <a href="https://arxiv.org/abs/2001.04743">Nonstandard Cayley automatic representations of fundamental groups</a>, arXiv:2001.04743 [math.GR], 2020.
%H A031396 Dmitry Berdinsky and Prohrak Kruengthomya, <a href="https://doi.org/10.1007/978-3-030-40608-0_7">Nonstandard Cayley Automatic Representations for Fundamental Groups of Torus Bundles over the Circle</a>, International Conference on Language and Automata Theory and Applications (LATA 2020): Language and Automata Theory and Applications, Lecture Notes in Computer Science, Vol 12038. Springer, Cham, 115-127.
%H A031396 Hsin-Te Chiang, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu, and Chiun-Chang Lee, <a href="https://doi.org/10.7546/nntdm.2018.24.3.10-26">On negative Pell equations: Solvability and unsolvability in integers</a>, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 10-26.
%H A031396 S. R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H A031396 D. Khurana and T. Y. Lam, <a href="http://dx.doi.org/10.1142/S0219498811004422">Invertible commutators in matrix rings</a>, J. Algebra and Applications, 10 (211), 51-71.
%H A031396 K. Lakshmi and R. Someshwari, <a href="https://www.ijeter.everscience.org/Manuscripts/Volume-4/Issue-7/Vol-4-issue-7-M-02.pdf">On The Negative Pell Equation y^2 = 72x^2 - 23</a>, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
%H A031396 Morris Newman, <a href="https://www.jstor.org/stable/2319968">A note on an equation related to the Pell equation</a>, The American Mathematical Monthly 84.5 (1977): 365-366.
%H A031396 R. Suganya and D. Maheswari, <a href="http://dx.doi.org/10.22457/jmi.v11a9">On the Negative Pellian Equation y^2 = 110 * x^2 - 29</a>, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
%H A031396 A. Vijayasankar, M. A. Gopalan, and V. Krithika, <a href="https://www.neliti.com/publications/257422/on-the-negative-pell-equation">On The Negative Pell Equation y^2 = 112 * x^2 - 7</a>, International Journal of Engineering and Applied Sciences (IJEAS 2017), Vol. 4, Issue 7, 11-14.
%t A031396 fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] != {}; Select[ Range@ 300, fQ] (* _Robert G. Wilson v_, Dec 19 2013 *)
%o A031396 (Sage)
%o A031396 def is_A031396(k):
%o A031396     if k==1: return True
%o A031396     if Integer(k).is_square(): return False
%o A031396     K.<a> = QuadraticField(k)
%o A031396     return continued_fraction(a).period_length()%2
%o A031396 print([k for k in range(1, 1000) if is_A031396(k)])  # _Robin Visser_, Nov 02 2024
%Y A031396 Equals {1} U A003814.
%Y A031396 Cf. A031398, A002313, A133204, A130226 (values of x).
%Y A031396 See also A322781, A323271, A323272.
%K A031396 nonn
%O A031396 1,2
%A A031396 _David W. Wilson_