A322781 -id:A322781 - 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

A323271 Numbers of the form pqr where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.

Original entry on oeis.org

2405, 3145, 4745, 6205, 6305, 8245, 8905, 9605, 12545, 12805, 14705, 16405, 16745, 17945, 18241, 19045, 19345, 19805, 20213, 20605, 20905, 22945, 23545, 25805, 26605, 26945, 28645, 29705, 30073, 33745, 35705, 35989, 36205, 36305, 37505, 38369, 38545

Offset: 1

N. J. A. Sloane, Jan 11 2019

nonn

If k is a term, the Pell equation x^2 - k*y^2 = -1 has a solution [Dirichlet, Newman (1977)]. This is only a sufficient condition, there are many other solutions, see A031396.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Morris Newman, A note on an equation related to the Pell equation, The American Mathematical Monthly 84.5 (1977): 365-366.

Cf. A002144, A031396, A322781, A323272.

from sympy.ntheory import legendre_symbol, factorint
A323271_list, k = [], 1
while len(A323271_list) < 10000:
    fk, fv = zip(*list(factorint(4*k+1).items()))
    if sum(fv) == len(fk) == 3 and fk[0] % 4 == fk[1] % 4 == fk[2] % 4 == 1 and legendre_symbol(fk[0],fk[1]) == legendre_symbol(fk[0],fk[2]) == legendre_symbol(fk[1],fk[2]) == -1:
            A323271_list.append(4*k+1)
    k += 1 # Chai Wah Wu, Jan 11 2019

A323272 Numbers of the form p_1p_2p_3...p_r where r is 2 or an odd number > 2, and the p_i are distinct primes congruent to 1 mod 4 such that Legendre(p_i/p_j) = -1 for all i != j.

Original entry on oeis.org

65, 85, 185, 265, 365, 481, 485, 493, 533, 565, 629, 685, 697, 785, 865, 949, 965, 985, 1037, 1073, 1157, 1165, 1189, 1241, 1261, 1285, 1385, 1417, 1465, 1565, 1585, 1649, 1685, 1765, 1769, 1781, 1853, 1865, 1921, 1937, 1985, 2117, 2165, 2173

Offset: 1

N. J. A. Sloane, Jan 11 2019

nonn

If k is a term, the Pell equation x^2 - k*y^2 = -1 has a solution [Dirichlet, Newman (1977)]. This is only a sufficient condition, there are many other solutions, see A031396.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Morris Newman, A note on an equation related to the Pell equation, The American Mathematical Monthly 84.5 (1977): 365-366.

Cf. A002144, A031396. Includes the union of A322781 and A323271.

cp's OEIS Frontend

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

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A323271 Numbers of the form pqr where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.

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A323272 Numbers of the form p_1p_2p_3...p_r where r is 2 or an odd number > 2, and the p_i are distinct primes congruent to 1 mod 4 such that Legendre(p_i/p_j) = -1 for all i != j.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Sage

A323271 Numbers of the form p*q*r where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

A323272 Numbers of the form p_1*p_2*p_3*...*p_r where r is 2 or an odd number > 2, and the p_i are distinct primes congruent to 1 mod 4 such that Legendre(p_i/p_j) = -1 for all i != j.

Views

Author

Keywords

Comments

Links

Crossrefs

A323271 Numbers of the form pqr where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.

A323272 Numbers of the form p_1p_2p_3...p_r where r is 2 or an odd number > 2, and the p_i are distinct primes congruent to 1 mod 4 such that Legendre(p_i/p_j) = -1 for all i != j.