A371957 Positive integers k such that the parametric Pell-type equation x^2 - m*x*y + y^2 = - m^2 - k has no integer solutions (x,y) for all integers m >= 1, excluding the cases k==1 (mod 4), k==3 (mod 9), and k==6 (mod 9).

4, 8, 14, 18, 19, 26, 38, 44, 47, 54, 63, 68, 74, 79, 98, 99, 103, 110, 118, 119, 124, 126, 134, 143, 144, 154, 158, 166, 179, 180, 194, 198, 199, 206, 207, 208, 214, 215, 224, 234, 238, 239, 250, 254, 263, 274, 278, 279, 287, 299, 306, 308, 314, 319, 324, 326, 334, 342, 351, 359, 362, 368, 374, 378, 383, 404, 406, 414, 418

Offset

1,1

Comments

For k == 1 (mod 4), k == 3 (mod 9), or k == 6 (mod 9), the equation x^2 - m*x*y + y^2 = - m^2 - k (*) has no integer solutions modulo 4 or 9. For other positive integer k, it suffice to check that the equation (*) has no integer solutions (x,y) for all integers m with 3<=m<=2*k+8 (see references for the proof of similar assertions). This condition can be verified by an algorithm similar to brute-force search for the general Pell equation x^2 - Dy^2 = N (see, for example, sect. 4.4.5 in: Andreescu T., Andrica D. Quadratic Diophantine Equations. New York: Springer, 2015).

For any other positive integer k, the equation (*) has integer solutions (x,y) for infinitely many integers m >= 1.

References

N. Osipov, A Pell-Type Diophantine Equation, Amer. Math. Monthly, 128 (2021), p. 858-860.

N. Osipov, A Pell-type Equation in Disguise, Amer. Math. Monthly, 129 (2022), p. 389-390.

Links

Table of n, a(n) for n=1..69.

Orlov Nikita, Pascal program.

Maple

check:=proc(k) local flag, m, y, mm, yy; flag:=0;
for m from 3 to 2*k+8 while flag=0 do
for y from 1 to evalf(sqrt((m^2+k)/(m-2)))+1 while flag=0 do
if issqr((m^2-4)*y^2-4*(m^2+k))=true then flag:=1; mm:=m; yy:=y; fi; od; od;
if flag=0 then return 0 else return [mm, yy]; fi; end proc:
for k from 1 to 2000 do
if k mod 4<>1 and k mod 9<>3 and k mod 9<>6 and check(k)=0 then print(k); fi; od:

Prog

(Pascal) // see link

Crossrefs

Cf. A370721 (for the equation x^2-m*x*y+y^2=m^2+k)

Sequence in context: A333465 A342274 A312401 * A312402 A312403 A312404

Adjacent sequences: A371954 A371955 A371956 * A371958 A371959 A371960

Keyword

nonn

Author

Orlov Nikita and Nikolay Osipov, Apr 14 2024

Extensions

Edited by Nikolay Osipov, Jun 11 2024

Status

approved