A370721 Positive integers k == 2 (mod 4) 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 integer m >= 1.

14, 94, 114, 118, 154, 158, 214, 238, 254, 294, 358, 414, 478, 574, 594, 598, 614, 654, 658, 694, 718, 758, 790, 814, 834, 862, 874, 878, 934, 958, 994, 1014, 1054, 1106, 1174, 1198, 1294, 1414, 1434, 1454, 1486, 1494, 1498, 1558, 1634, 1678, 1738, 1774, 1794, 1834, 1894, 1918, 1978

Offset: 1

Author

Orlov Nikita and Nikolay Osipov, Mar 07 2024

Keywords

• nonn

Comments

For a positive integer k == 2 (mod 4), it suffice to check that the equation x^2-m*x*y+y^2 = m^2+k (*) has no integer solutions (x,y) for all integer m with 1 <= m <= k/2 (see references for the proof of some 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).

Also, the equation (*) has no integer solutions (x,y) for all integer m >= 1 when k = 1 or k = 4. 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. A371957 (for the equation x^2-m*x*y+y^2=-m^2-k).

Programs

Maple

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

Pascal

(* see link *)

Extensions

Edited by Nikolay Osipov, Jun 11 2024