A031398 Squarefree n with no 4k+3 factors such that Pell equation x^2 - n 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

Author

David W. Wilson

Keywords

• nonn

Comments

Or, numbers n which are the sum of two relatively-prime squares but for which x^2 - n*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, 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, 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, K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) 4:2 (2016).

Crossrefs

Cf. A031396, A031397.

Programs

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 *)

Extensions

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