Orlov Nikita - cp's OEIS frontend

User: Orlov Nikita

Orlov Nikita has authored 2 sequences.

A371957 Positive integers k such that the parametric Pell-type equation x^2 - mxy + 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

Orlov Nikita and Nikolay Osipov, Apr 14 2024

nonn

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.

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.

Orlov Nikita, Pascal program.

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

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
```

Edited by Nikolay Osipov, Jun 11 2024

A370721 Positive integers k == 2 (mod 4) such that the parametric Pell-type equation x^2 - mxy + y^2 = m^2 + k has no integer solutions (x,y) for all integer m >= 1.

Original entry on oeis.org

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

Orlov Nikita and Nikolay Osipov, Mar 07 2024

nonn

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.

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.

Orlov Nikita, Pascal program.

Cf. A371957 (for the equation x^2-m*x*y+y^2=-m^2-k).

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 *)
```

Edited by Nikolay Osipov, Jun 11 2024

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User: Orlov Nikita

A371957 Positive integers k such that the parametric Pell-type equation x^2 - mxy + 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).

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A370721 Positive integers k == 2 (mod 4) such that the parametric Pell-type equation x^2 - mxy + y^2 = m^2 + k has no integer solutions (x,y) for all integer m >= 1.

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cp's OEIS Frontend

User: Orlov Nikita

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).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Pascal

Extensions

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Pascal

Extensions

A371957 Positive integers k such that the parametric Pell-type equation x^2 - mxy + 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).

A370721 Positive integers k == 2 (mod 4) such that the parametric Pell-type equation x^2 - mxy + y^2 = m^2 + k has no integer solutions (x,y) for all integer m >= 1.