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

1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 290, 293, 298

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

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

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

References

• Harvey Cohn, "Advanced Number Theory".

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Dmitry Berdinsky and Prohrak Kruengthomya, Nonstandard Cayley automatic representations of fundamental groups, arXiv:2001.04743 [math.GR], 2020.
• Dmitry Berdinsky and Prohrak Kruengthomya, Nonstandard Cayley Automatic Representations for Fundamental Groups of Torus Bundles over the Circle, 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.
• Hsin-Te Chiang, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu, and Chiun-Chang Lee, On negative Pell equations: Solvability and unsolvability in integers, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 10-26.
• S. R. Finch, Class number theory [Cached copy, with permission of the author]
• D. Khurana and T. Y. Lam, Invertible commutators in matrix rings, J. Algebra and Applications, 10 (211), 51-71.
• K. Lakshmi and R. Someshwari, On The Negative Pell Equation y^2 = 72x^2 - 23, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
• Morris Newman, A note on an equation related to the Pell equation, The American Mathematical Monthly 84.5 (1977): 365-366.
• R. Suganya and D. Maheswari, On the Negative Pellian Equation y^2 = 110 * x^2 - 29, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
• A. Vijayasankar, M. A. Gopalan, and V. Krithika, On The Negative Pell Equation y^2 = 112 * x^2 - 7, International Journal of Engineering and Applied Sciences (IJEAS 2017), Vol. 4, Issue 7, 11-14.

Crossrefs

Equals {1} U A003814.

Cf. A031398, A002313, A133204, A130226 (values of x).

See also A322781, A323271, A323272.

Programs

Mathematica

fQ[n_] := Solve[x^2 + 1 == n*y^2, {x, y}, Integers] != {}; Select[ Range@ 300, fQ] (* Robert G. Wilson v, Dec 19 2013 *)

Sage

def is_A031396(k):
    if k==1: return True
    if Integer(k).is_square(): return False
    K. = QuadraticField(k)
    return continued_fraction(a).period_length()%2
print([k for k in range(1, 1000) if is_A031396(k)])  # Robin Visser, Nov 02 2024