A031398 -id:A031398 - 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

A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

Original entry on oeis.org

2, 5, 10, 13, 17, 26, 29, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 313, 314, 317, 337, 346, 349, 353, 362

Offset: 1

N. J. A. Sloane, Mira Bernstein. Entry revised by N. J. A. Sloane, Jun 11 2012

nonn

The squarefree elements of A003814 and A172000. - Max Alekseyev, Jun 01 2009
Together with {1} and A031398 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010
Squarefree integers m such that Q(sqrt(m)) contains the infinite continued fraction [k, k, k, k, k, ...] for some positive integer k. For example, Q(sqrt(5)) contains [1, 1, 1, 1, 1, ...] which equals (1 + sqrt(5))/2. - Greg Dresden, Jul 23 2010

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 46.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 56.
W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 1..9446
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
P. Seeling, Ueber die Aufloesung der Gleichung x^2-Ay^2=+-1 in ganzen Zahlen, wo A positiv und kein vollstaendiges Quadrat sein muss, Archiv der Mathematik und Physik, Vol. 52 (1871), p. 40-49.

Cf. A031396, A031397, A003814, A249021.

isA003654 := proc(n)
    local cf,p ;
    if not numtheory[issqrfree](n) then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 3 then
            return false;
        end if;
    end do:
    cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;
    type( nops(op(2,cf)),'odd') ;
end proc:
A003654 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA003654(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003654(n),n=1..40) ; # R. J. Mathar, Oct 19 2014

Reap[For[n = 2, n < 1000, n++, If[SquareFreeQ[n], sol = Solve[x^2 - n y^2 == -1, {x, y}, Integers]; If[sol != {}, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Mar 24 2020 *)

Edited by Max Alekseyev, Mar 17 2010

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 146, 149, 157, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 305, 313, 314, 317, 337, 346, 349

Offset: 1

Steven Finch

nonn

Primitively but not imprimitively represented by x^2 + y^2.
The disjoint union of {1}, A003654, and A031398. - Max Alekseyev, Mar 09 2010
Squarefree members of A202057. - Artur Jasinski, Dec 10 2011
Union of A231754 and 2*A231754. Squarefree numbers whose prime factors are in A002313. - Robert Israel, Aug 23 2017
It appears that a(n) is the n-th index, k, such that f(k) = 2, where f(k) = 3*(Sum_{i=1..k} floor(i^2/k)) - k^2 (see A175908). - John W. Layman, May 16 2011

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Index entries for sequences related to sums of squares

Cf. A001481, A008784, A022544, A034023, A175908.

Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

a020893 n = a020893_list !! (n-1)
a020893_list = filter (\x -> any (== 1) $ map (a010052 . (x -)) $
                             takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, May 28 2015

N:= 1000: # to get all terms <= N
R:= {1,2}:
p:= 2:
do
p:= nextprime(p);
if p > N then break fi;
if p mod 4 <> 1 then next fi;
R:= R union select(`<=`,map(`*`,R,p),N);
od:
sort(convert(R,list)); # Robert Israel, Aug 23 2017

lim = 17; t = Join[{1}, Select[Union[Flatten[Table[x^2 + y^2, {x, lim}, {y, x}]]], # < lim^2 && SquareFreeQ[#] &]]
Select[Union[Total/@Tuples[Range[0,20]^2,2]],SquareFreeQ] (* Harvey P. Dale, Jul 26 2017 *)
Block[{nn = 350, p}, p = {1, 2}~Join~Select[Prime@ Range@ PrimePi@ nn, Mod[#, 4] == 1 &]; Select[Range@ nn, And[SquareFreeQ@ #, SubsetQ[p, FactorInteger[#][[All, 1]]]] &]] (* Michael De Vlieger, Aug 23 2017 *)
(* or *)
Select[Range[350], SquareFreeQ@ # && ! MemberQ[Mod[First /@ FactorInteger@ #, 4], 3] &] (* Giovanni Resta, Aug 25 2017 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 || f[i,1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015

from itertools import count, islice
from sympy import factorint
def A020893_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 and e == 1 for p, e in factorint(n).items()),count(1))
A020893_list = list(islice(A020893_gen(),30)) # Chai Wah Wu, Jun 28 2022

a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015

Edited by N. J. A. Sloane, Aug 30 2017

A031399 Numbers n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 insoluble.

Original entry on oeis.org

4, 8, 16, 20, 25, 32, 34, 40, 52, 64, 68, 80, 100, 104, 116, 128, 136, 146, 148, 160, 164, 169, 178, 194, 200, 205, 208, 212, 221, 232, 244, 256, 260, 272, 289, 292, 296, 305, 320, 328, 340, 356, 377, 386, 388, 400, 404, 410, 416, 424, 436, 452

Offset: 1

David W. Wilson

nonn

"Advanced Number Theory" by Harvey Cohn.

Joerg Arndt, Matters Computational (The Fxtbook)

Cf. A031396, A031397, A031398.

A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

Original entry on oeis.org

34, 136, 146, 178, 194, 205, 221, 305, 306, 377, 386, 410, 466, 482, 505, 514, 544, 545, 562, 584, 674, 689, 706, 712, 745, 776, 793, 802, 820, 850, 866, 884, 890, 898, 905, 1154, 1186, 1202, 1205, 1220, 1224, 1234, 1282, 1314, 1345, 1346, 1394, 1405, 1469

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

If the fundamental unit y0 + x0*sqrt(n) of Q(sqrt(n)) has norm -1, then (x0,y0) represents a rational solution to Pell equation y^2 - n*x^2 = -1. For n in this sequence, rational solutions exist but not delivered by the fundamental unit.

Set difference of A000415 and its subsequence A172000.
Set difference of A087643 and its subsequence A022544.
Squarefree terms form A031398.
Odd terms form A249052.
Cf. A031399, A137351.

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398.

Edited by Max Alekseyev, Mar 09 2010

A249052 Odd numbers n without prime factors of the form 4k+3 such that x^2-n*y^2=-1 has no solutions.

Original entry on oeis.org

205, 221, 305, 377, 505, 545, 689, 725, 745, 793, 905, 1205, 1345, 1405, 1469, 1513, 1517, 1537, 1717, 1885, 1945, 1961, 2005, 2041, 2045, 2105, 2225, 2245, 2329, 2353, 2525, 2533, 2669, 2701, 2845, 2993, 3005, 3205, 3305, 3497, 3505, 3737, 3757, 3805, 3893, 3965, 4069, 4105, 4145, 4205, 4321, 4369

Offset: 1

Joerg Arndt, Apr 09 2008

nonn

A positive odd integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398. - Max Alekseyev, Jun 26 2022

Odd terms of A172001.

Cf. A031398, A031399, A137351.

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

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A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

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A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

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A031399 Numbers n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 insoluble.

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A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

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A249052 Odd numbers n without prime factors of the form 4k+3 such that x^2-n*y^2=-1 has no solutions.

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