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