A130226 -id:A130226 - 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

A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

Original entry on oeis.org

2, 18, 4, 70, 6, 32, 182, 29718, 1068, 500, 5604, 10, 8890182, 776, 1744, 113582, 4832118, 1118, 1111225770, 1764132, 14, 1710, 23156, 71011068, 16, 82, 8920484118, 1063532, 2482, 126862368, 352618

Offset: 1

Matthijs Coster, Apr 29 2004

nonn

Subsequence of A191860. [Reinhard Zumkeller, Jun 18 2011]

Cf. A002144, A094049 (associated k), A130226, A137351, A179073.

Cf. A000196, A037213, A000290.

a094048 n = head [m | m <- map (a037213 . subtract 1 . (* a002144 n))
                               (tail a000290_list), m > 0]
-- Reinhard Zumkeller, Jun 13 2015

f[n_] := Block[{y = 1}, While[ !IntegerQ[ Sqrt[n*y^2 - 1]], y++ ]; Sqrt[n*y^2 - 1]]; lst = {}; Do[p = Prime@ n; If[ Mod[p, 4] == 1, AppendTo[lst, f@p]; Print[{n, Prime@n, f@p}]], {n, 66}]; lst

Edited by Don Reble, Apr 30 2004

A130227 Smallest integer y satisfying the Pell equation x^2-ny^2=-1 for the values of n given in A031396.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 1, 5, 1, 25, 13, 3805, 1, 125, 5, 1, 41, 53, 569, 1, 389, 851525, 73, 1, 61, 5, 149, 1, 9305, 385645, 1, 85, 82596761, 5, 126985, 1, 221, 17, 1, 113, 1517, 4574225, 281

Offset: 1

Colin Barker, Aug 05 2007

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A031396, A130226.

A249021 Value x in the solution of x^2-D*y^2=-1 as D runs through A003654.

Original entry on oeis.org

7, 38, 117, 18, 268, 515, 70, 882, 32, 182, 99, 29718, 2072, 1068, 43, 2943, 378, 500, 5604, 4030, 4005, 8890182, 776, 5357, 57, 1744, 6948, 113582, 4832118, 8827, 1118, 1111225770, 68, 1764132, 11018, 3141, 251, 13545, 1710, 23156, 71011068, 16432, 6072, 82, 1407, 8920484118, 1063532, 19703

Offset: 1

R. J. Mathar, Oct 19 2014

nonn

The pair (x,y) is taken from the numerator of the earliest (lowest order) convergent to the continued fraction of sqrt(D) that satisfies the "non-Pell" equation.

Cf. A130226.

A249021 := proc(n)
    local dis,cf,o,q,x,y ;
    dis := A003654(n) ;
    cf := numtheory[cfrac](sqrt(dis),'periodic','quotients') ;
    for o from 1 do
        q := numtheory[nthconver](cf,o) ;
        x := numer(q) ;
        y := denom(q) ;
        if x^2-dis*y^2 = -1 then
            return x ;
        end if;
    end do:
end proc:
seq(A249021(n),n=1..50) ;

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

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A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

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A130227 Smallest integer y satisfying the Pell equation x^2-ny^2=-1 for the values of n given in A031396.

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A249021 Value x in the solution of x^2-D*y^2=-1 as D runs through A003654.

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Maple