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

Author

Orlov Nikita and Nikolay Osipov, Apr 14 2024

Keywords

• nonn

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.

Crossrefs

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

Programs

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:

Pascal

// see link

Extensions

Edited by Nikolay Osipov, Jun 11 2024