A133204 -id:A133204 - cp's OEIS frontend

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

David W. Wilson

nonn

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

Harvey Cohn, "Advanced Number Theory".

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.

Equals {1} U A003814.
Cf. A031398, A002313, A133204, A130226 (values of x).
See also A322781, A323271, A323272.

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

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

A385224 Primes p such that multiplicative order of -4 modulo p is odd.

Original entry on oeis.org

5, 13, 29, 37, 41, 53, 61, 101, 109, 113, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 313, 317, 349, 373, 389, 397, 409, 421, 457, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 761, 773, 797, 809, 821, 829, 853, 857, 877, 941, 953, 997

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of -4 modulo a(n) is A385230(n).
Different from A133204: 593 is here but not in A133204, and 1601 is in A133204 but not here.
The sequence contains no primes congruent to 3 modulo 4 and all primes congruent to 5 modulo 8:
  - If p is a term of this sequence, then -4 is a quadratic residue modulo p, so p == 1 (mod 4);
  - For p == 1 (mod 4), we have (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i is a solution to i^2 == -1 (mod p).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A002144 (primes congruent to 1 modulo 4).
Contains A007521 (primes congruent to 5 or modulo 8) as a proper subsequence.
Cf. A385230 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), this sequence (base -4), A385225 (base -5).
Cf. A133204.

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2

cp's OEIS Frontend

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

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