A031398 Squarefree m with no 4k+3 factors such that Pell equation x^2 - m*y^2 = -1 is insoluble.

34, 146, 178, 194, 205, 221, 305, 377, 386, 410, 466, 482, 505, 514, 545, 562, 674, 689, 706, 745, 793, 802, 866, 890, 898, 905, 1154, 1186, 1202, 1205, 1234, 1282, 1345, 1346, 1394, 1405, 1469, 1513, 1517, 1537, 1538, 1717, 1762, 1802, 1858

Offset

1,1

Comments

Or, numbers m which are the sum of two relatively-prime squares but for which x^2 - m*y^2 does not represent -1.

Together with {1} and A003654 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010

References

Harvey Cohn, "Advanced Number Theory".

Links

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Janis Kuzmanis, A simple solvability criterion for the negative Pell equation, hal-02502164, Mathematics [math] / Number Theory [math.NT], (2020).

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).

J. P. Robertson and K. R. Matthews, A continued fraction approach to a result of Feit, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.

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.

S. Vidhyalakshmi, V. Krithika, and K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) 4:2 (2016).

Mathematica

sel = Select[Range[2000], SquareFreeQ[#] && FreeQ[Mod[FactorInteger[#][[All, 1]], 4], 3]&]; r[n_] := Reduce[x^2-n*y^2 == -1, {x, y}, Integers]; Reap[For[n=1, n <= Length[sel], n++, an = sel[[n]]; If[r[an] === False, Print[an]; Sow[an]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2014 *)

Prog

(Python)
from itertools import count, islice
from sympy import factorint
from sympy.solvers.diophantine.diophantine import diop_DN
def A031398_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: max((f:=factorint(n)).values(), default=0)<2 and not (any(p&3==3 for p in f) or len(diop_DN(n, -1))), count(max(startvalue, 1)))
A031398_list = list(islice(A031398_gen(), 50)) # Chai Wah Wu, Dec 22 2025

Crossrefs

Cf. A031396, A031397.

Sequence in context: A280550 A105714 A072319 * A259954 A180759 A159744

Adjacent sequences: A031395 A031396 A031397 * A031399 A031400 A031401

Keyword

nonn

Author

David W. Wilson

Extensions

Edited by N. J. A. Sloane, Apr 28 2008, at the suggestion of Artur Jasinski

Status

approved