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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).
2
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.
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
KEYWORD
nonn
AUTHOR
Orlov Nikita and Nikolay Osipov, Apr 14 2024
EXTENSIONS
Edited by Nikolay Osipov, Jun 11 2024
STATUS
approved