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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).

%I A371957 #28 Jun 12 2024 09:45:18
%S A371957 4,8,14,18,19,26,38,44,47,54,63,68,74,79,98,99,103,110,118,119,124,
%T A371957 126,134,143,144,154,158,166,179,180,194,198,199,206,207,208,214,215,
%U A371957 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
%N 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).
%C A371957 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).
%C A371957 For any other positive integer k, the equation (*) has integer solutions (x,y) for infinitely many integers m >= 1.
%D A371957 N. Osipov, A Pell-Type Diophantine Equation, Amer. Math. Monthly, 128 (2021), p. 858-860.
%D A371957 N. Osipov, A Pell-type Equation in Disguise, Amer. Math. Monthly, 129 (2022), p. 389-390.
%H A371957 Orlov Nikita, <a href="/A371957/a371957.pas.txt">Pascal program</a>.
%p A371957 check:=proc(k) local flag,m,y,mm,yy; flag:=0;
%p A371957 for m from 3 to 2*k+8 while flag=0 do
%p A371957 for y from 1 to evalf(sqrt((m^2+k)/(m-2)))+1 while flag=0 do
%p A371957 if issqr((m^2-4)*y^2-4*(m^2+k))=true then flag:=1; mm:=m; yy:=y; fi; od; od;
%p A371957 if flag=0 then return 0 else return [mm,yy]; fi; end proc:
%p A371957 for k from 1 to 2000 do
%p A371957 if k mod 4<>1 and k mod 9<>3 and k mod 9<>6 and check(k)=0 then print(k); fi; od:
%o A371957 (Pascal) // see link
%Y A371957 Cf. A370721 (for the equation x^2-m*x*y+y^2=m^2+k)
%K A371957 nonn
%O A371957 1,1
%A A371957 _Orlov Nikita_ and _Nikolay Osipov_, Apr 14 2024
%E A371957 Edited by _Nikolay Osipov_, Jun 11 2024