A031396 -id:A031396 - cp's OEIS frontend

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

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.

A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

Original entry on oeis.org

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 39, 43, 52, 57, 61, 67, 73, 76, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 151, 157, 163, 172, 181, 183, 193, 199, 201, 211, 217, 228, 237, 241, 244, 247, 259, 268, 271, 273, 277, 283, 292, 301, 307, 309, 313, 327, 331, 337, 343, 364

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [1, 2]], [3, [0, 1]], [4, [1, 1]], [7, [2, 1]], [12, [3, 1]], [13, [7, 2]], [19, [4, 1]], [21, [9, 2]], [28, [5, 1]], [31, [11, 2]], [39, [6, 1]], [43, [13, 2]], [52, [7, 1]], [57, [15, 2]], [61, [5639, 722]], [67, [8, 1]], [73, [17, 2]], [76, [61, 7]], [84, [9, 1]], [91, [19, 2]], [93, [135, 14]], [97, [847, 86]], [103, [10, 1]], [109, [1399, 134]], [111, [21, 2]], [124, [11, 1]], [127, [293, 26]], [129, [159, 14]], [133, [23, 2]], [139, [224, 19]], [147, [12, 1]], [151, [86, 7]], [157, [25, 2]], [163, [932, 73]], [172, [13, 1]], [181, [11262809, 837158]], [183, [27, 2]], [193, [189743, 13658]], [199, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A244819, A261246, A377607.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377600_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,-3)), count(max(startvalue,1)))
A377600_list = list(islice(A377600_gen(),63)) # Chai Wah Wu, Nov 03 2024

A377607 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 3 is solvable over the integers.

Original entry on oeis.org

1, 6, 13, 22, 33, 46, 61, 69, 73, 78, 94, 97, 109, 118, 141, 157, 166, 177, 181, 193, 213, 214, 222, 241, 249, 253, 262, 277, 286, 313, 321, 334, 337, 358, 366, 382, 393, 397, 409, 421, 429, 433, 438, 454, 457, 478, 481, 501, 502, 517, 526, 537, 541, 573, 598, 601, 613, 622, 649, 654, 661

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [2, 1]], [6, [3, 1]], [13, [4, 1]], [22, [5, 1]], [33, [6, 1]], [46, [7, 1]], [61, [8, 1]], [69, [108, 13]], [73, [94, 11]], [78, [9, 1]], [94, [223, 23]], [97, [10, 1]], [109, [9532, 913]], [118, [11, 1]], [141, [12, 1]], [157, [289580, 23111]], [166, [13, 1]], [177, [306, 23]], [181, [148, 11]], [193, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A243655, A261246, A377600.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377607_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,3)), count(max(startvalue,1)))
A377607_list = list(islice(A377607_gen(),61)) # Chai Wah Wu, Nov 03 2024

A383734 Numbers k such that 2+k and 2*k are squares.

Original entry on oeis.org

2, 98, 3362, 114242, 3880898, 131836322, 4478554082, 152139002498, 5168247530882, 175568277047522, 5964153172084898, 202605639573839042, 6882627592338442562, 233806732499933208098, 7942546277405390632802, 269812766699283348307202, 9165691521498228451812098

Offset: 1

Emilio Martín, May 07 2025

The limit of a(n+1)/a(n) is 33.97056... = 17+12*sqrt(2) = (3+2*sqrt(2))^2 (see A156164).

			98 is a term becouse 98+2=100 is a square and 98*2=196 is a square.

Eric Weisstein's World of Mathematics, NSW numbers.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A002315, A088165, A008843, A008844, A075870, A031396, A156164.
Cf. A382209 (10+k and 10*k are squares).
Cf. A245226 (m such that k+m and k*m are squares).

LinearRecurrence[{35, -35, 1}, {2, 98, 3362}, 20] (* Amiram Eldar, May 07 2025 *)

from itertools import islice
def A383734_gen(): # generator of terms
    x, y = 1, 7
    while True:
        yield 2*x**2
        x, y = y, 6*y - x
A383734_list = list(islice(A383734_gen(), 100))

a(n) = (1/2) * ((3+2*sqrt(2))^(2*n-1) + (3-2*sqrt(2))^(1-2*n)) - 1.
a(n) = -2*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 3*cosh(n*log(17+12*sqrt(2))) - 1.
a(n) = 2*A002315(n-1)^2.
a(n) = A075870(n)^2 - 2.
a(n) = 34*a(n-1) - a(n-2) + 32.
G.f.: 2 * (1 + 14*x + x^2) / ((1 - x)*(1 - 34*x + x^2)). - Stefano Spezia, May 08 2025

A240950 Numbers k such that the continued fraction for sqrt(k) has odd period, omitting those values of k of the form m^2+1.

Original entry on oeis.org

13, 29, 41, 53, 58, 61, 73, 74, 85, 89, 97, 106, 109, 113, 125, 130, 137, 149, 157, 173, 181, 185, 193, 202, 218, 229, 233, 241, 250, 265, 269, 274, 277, 281, 293, 298, 313, 314, 317, 337, 338, 346, 349, 353, 365, 370, 373, 389, 394, 397, 409, 421, 425, 433

Offset: 1

Takao Ito, Aug 04 2014

nonn

p^2 - n*q^2 = -1 is solvable for integers p and q.

A031396 is the union of this sequence with A002522.

			sqrt(13) = [3;1,1,1,1,6].
sqrt(29) = [5;2,1,1,2,10].

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A002522, A003814, A031396.

More terms from Colin Barker, Dec 18 2014

A282341 Primes p of the form x^2 + y^2 such that q = (x^2 + 1)/y^2 is a prime less than p.

Original entry on oeis.org

349, 1049, 1733, 33749, 53849, 79549, 135449, 381949, 535849, 558149, 692249, 1036349, 1156249, 1483549, 1871449, 2304349, 3097769, 6181349, 6411049, 8809049, 10355549, 11102249, 16401701, 16491521, 22867549, 26419769, 27457889, 30603049, 31728577, 34176557

Offset: 1

Thomas Ordowski and Altug Alkan, Feb 12 2017

nonn

The negative Pell equation x^2 - q*y^2 = -1, hence q = (x^2 + 1)/y^2.

Primes p = q are A002496.

			For prime p = 349 = 18^2 + 5^2 is q = (18^2 + 1)/5^2 = 13 prime < p.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002313.

Cf. A002496, A031396.

list(lim)=my(v=List(),x2,q,y,p); for(x=1,sqrtint(lim\4), x2=4*x^2; [q,y]=core(x2+1,1); p=x2+y^2; if(qCharles R Greathouse IV, Feb 14 2017

A377598 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -2 is solvable over the integers.

Original entry on oeis.org

2, 3, 6, 11, 18, 19, 22, 27, 38, 43, 51, 54, 59, 66, 67, 83, 86, 102, 107, 114, 118, 123, 131, 134, 139, 146, 162, 163, 166, 171, 178, 179, 187, 198, 211, 214, 227, 242, 243, 246, 251, 258, 262, 267, 278, 283, 291, 307, 326, 331, 339, 347, 354, 358, 363, 374, 379, 387, 402, 411, 418, 419

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[2, [0, 1]], [3, [1, 1]], [6, [2, 1]], [11, [3, 1]], [18, [4, 1]], [19, [13, 3]], [22, [14, 3]], [27, [5, 1]], [38, [6, 1]], [43, [59, 9]], [51, [7, 1]], [54, [22, 3]], [59, [23, 3]], [66, [8, 1]], [67, [221, 27]], [83, [9, 1]], [86, [102, 11]], [102, [10, 1]], [107, [31, 3]], [114, [32, 3]], [118, [554, 51]], [123, [11, 1]], [131, [103, 9]], [134, [382, 33]], [139, [8807, 747]], [146, [12, 1]], [162, [140, 11]], [163, [8005, 627]], [166, [41242, 3201]], [171, [13, 1]], [178, [40, 3]], [179, [2047, 153]], [187, [41, 3]], [198, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A261246.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377598_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,-2)), count(max(startvalue,1)))
A377598_list = list(islice(A377598_gen(),62)) # Chai Wah Wu, Nov 03 2024

A356488 Numbers k such that the equation x^2 - k*y^4 = -1 has a solution for which |y| > 2.

Original entry on oeis.org

2, 53, 314, 1042, 1685, 1825, 3281, 4586, 5521, 6770, 8597, 9050, 11509, 13858, 17498, 20369, 24737, 28085, 28130, 29041, 31226, 33226, 37141, 37585, 42965, 47402, 49205, 53954, 57125, 58913, 66193, 71674, 79682, 85685, 94421, 100946, 110410, 113290, 115202

Offset: 1

Jinyuan Wang, Aug 09 2022

nonn

For k > 2, the equation x^2 - k*y^4 = -1 has at most one positive integer solution. If this solution (x, y) exists, we have v = y^2, where v is the smallest integer satisfying the Pell equation u^2 - k*v^2 = -1 (A130227).

			The equation x^2 - 2*y^4 = -1 has only two positive solutions (1, 1) and (239, 13), so 2 is in the sequence.

Chen Jian Hua and Paul Voutier, Complete solution of the diophantine equation X^2 + 1 = dY^4 and a related family of quartic Thue equations, arXiv:1401.5450 [math.NT], 2014-2018.

Cf. A031396, A130227, A182468.

cp's OEIS Frontend

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

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A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

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A377607 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 3 is solvable over the integers.

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A383734 Numbers k such that 2+k and 2*k are squares.

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Mathematica

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A240950 Numbers k such that the continued fraction for sqrt(k) has odd period, omitting those values of k of the form m^2+1.

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A282341 Primes p of the form x^2 + y^2 such that q = (x^2 + 1)/y^2 is a prime less than p.

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PARI

A377598 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -2 is solvable over the integers.

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Programs

Python

A356488 Numbers k such that the equation x^2 - k*y^4 = -1 has a solution for which |y| > 2.

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