How to solve this Diophantine equation without solving each equation independently?

Question

Find all integral solutions to the equation $x^2 + 4xy - y^2 = m$ with $-5 \leq m \leq 10$.

I know that I can set $m = -5$ to $m = 10$ and solve all of the equations independently. But is there any better method to this question?

Answer 1

This Pell equation $(x + 2 y)^2 - 5 y^2 = m$.

Some solutions $(m,x,y)$:

(-5, 2, 9)
(-5, 38, 161)
(-5, 682, 2889)
(-5, 12238, 51841)
(-5, 219602, 930249)
(-5, 3940598, 16692641)
(-5, 70711162, 299537289)
(-5, 1268860318, 5374978561)
(-5, 22768774562, 96450076809)
(-5, 408569081798, 1730726404001)
(-5, 7331474697802, 31056625195209)
(-5, 131557975478638, 557288527109761)
(-5, 2360712083917682, 10000136862780489)
(-5, 42361259535039638, 179445175002939041)
(-5, 760141959546795802, 3220013013190122249)
(-5, 13640194012307284798, 57780789062419261441)
(-5, 244763350261984330562, 1036834190110356583689)
(-4, -1, 1)
(-4, 0, 2)
(-4, 1, 5)
(-4, 3, 13)
(-4, 8, 34)
(-4, 21, 89)
(-4, 55, 233)
(-4, 144, 610)
(-4, 377, 1597)
(-4, 987, 4181)
(-4, 2584, 10946)
(-4, 6765, 28657)
(-4, 17711, 75025)
(-4, 46368, 196418)
(-4, 121393, 514229)
(-4, 317811, 1346269)
(-4, 832040, 3524578)
(-4, 2178309, 9227465)
(-4, 5702887, 24157817)
(-4, 14930352, 63245986)
(-4, 39088169, 165580141)
(-4, 102334155, 433494437)
(-4, 267914296, 1134903170)
(-4, 701408733, 2971215073)
(-4, 1836311903, 7778742049)
(-4, 4807526976, 20365011074)
(-4, 12586269025, 53316291173)
(-4, 32951280099, 139583862445)
(-4, 86267571272, 365435296162)
(-4, 225851433717, 956722026041)
(-4, 591286729879, 2504730781961)
(-4, 1548008755920, 6557470319842)
(-4, 4052739537881, 17167680177565)
(-4, 10610209857723, 44945570212853)
(-4, 27777890035288, 117669030460994)
(-4, 72723460248141, 308061521170129)
(-4, 190392490709135, 806515533049393)
(-4, 498454011879264, 2111485077978050)
(-4, 1304969544928657, 5527939700884757)
(-4, 3416454622906707, 14472334024676221)
(-4, 8944394323791464, 37889062373143906)
(-4, 23416728348467685, 99194853094755497)
(-4, 61305790721611591, 259695496911122585)
(-4, 160500643816367088, 679891637638612258)
(-4, 420196140727489673, 1779979416004714189)
(-4, 1100087778366101931, 4660046610375530309)
(-4, 2880067194370816120, 12200160415121876738)
(-4, 7540113804746346429, 31940434634990099905)
(-4, 19740274219868223167, 83621143489848422977)
(-4, 51680708854858323072, 218922995834555169026)
(-4, 135301852344706746049, 573147844013817084101)
(-1, 0, 1)
(-1, 4, 17)
(-1, 72, 305)
(-1, 1292, 5473)
(-1, 23184, 98209)
(-1, 416020, 1762289)
(-1, 7465176, 31622993)
(-1, 133957148, 567451585)
(-1, 2403763488, 10182505537)
(-1, 43133785636, 182717648081)
(-1, 774004377960, 3278735159921)
(-1, 13888945017644, 58834515230497)
(-1, 249227005939632, 1055742538989025)
(-1, 4472197161895732, 18944531186571953)
(-1, 80250321908183544, 339945818819306129)
(-1, 1440033597185408060, 6100080207560938369)
(-1, 25840354427429161536, 109461497917277584513)
(1, 1, 4)
(1, 17, 72)
(1, 305, 1292)
(1, 5473, 23184)
(1, 98209, 416020)
(1, 1762289, 7465176)
(1, 31622993, 133957148)
(1, 567451585, 2403763488)
(1, 10182505537, 43133785636)
(1, 182717648081, 774004377960)
(1, 3278735159921, 13888945017644)
(1, 58834515230497, 249227005939632)
(1, 1055742538989025, 4472197161895732)
(1, 18944531186571953, 80250321908183544)
(1, 339945818819306129, 1440033597185408060)
(1, 6100080207560938369, 25840354427429161536)
(4, 1, 1)
(4, 1, 3)
(4, 2, 8)
(4, 5, 21)
(4, 13, 55)
(4, 34, 144)
(4, 89, 377)
(4, 233, 987)
(4, 610, 2584)
(4, 1597, 6765)
(4, 4181, 17711)
(4, 10946, 46368)
(4, 28657, 121393)
(4, 75025, 317811)
(4, 196418, 832040)
(4, 514229, 2178309)
(4, 1346269, 5702887)
(4, 3524578, 14930352)
(4, 9227465, 39088169)
(4, 24157817, 102334155)
(4, 63245986, 267914296)
(4, 165580141, 701408733)
(4, 433494437, 1836311903)
(4, 1134903170, 4807526976)
(4, 2971215073, 12586269025)
(4, 7778742049, 32951280099)
(4, 20365011074, 86267571272)
(4, 53316291173, 225851433717)
(4, 139583862445, 591286729879)
(4, 365435296162, 1548008755920)
(4, 956722026041, 4052739537881)
(4, 2504730781961, 10610209857723)
(4, 6557470319842, 27777890035288)
(4, 17167680177565, 72723460248141)
(4, 44945570212853, 190392490709135)
(4, 117669030460994, 498454011879264)
(4, 308061521170129, 1304969544928657)
(4, 806515533049393, 3416454622906707)
(4, 2111485077978050, 8944394323791464)
(4, 5527939700884757, 23416728348467685)
(4, 14472334024676221, 61305790721611591)
(4, 37889062373143906, 160500643816367088)
(4, 99194853094755497, 420196140727489673)
(4, 259695496911122585, 1100087778366101931)
(4, 679891637638612258, 2880067194370816120)
(4, 1779979416004714189, 7540113804746346429)
(4, 4660046610375530309, 19740274219868223167)
(4, 12200160415121876738, 51680708854858323072)
(4, 31940434634990099905, 135301852344706746049)
(4, 83621143489848422977, 354224848179261915075)
(5, 1, 2)
(5, 9, 38)
(5, 161, 682)
(5, 2889, 12238)
(5, 51841, 219602)
(5, 930249, 3940598)
(5, 16692641, 70711162)
(5, 299537289, 1268860318)
(5, 5374978561, 22768774562)
(5, 96450076809, 408569081798)
(5, 1730726404001, 7331474697802)
(5, 31056625195209, 131557975478638)
(5, 557288527109761, 2360712083917682)
(5, 10000136862780489, 42361259535039638)
(5, 179445175002939041, 760141959546795802)
(5, 3220013013190122249, 13640194012307284798)
(5, 57780789062419261441, 244763350261984330562)
(9, 3, 12)
(9, 51, 216)
(9, 915, 3876)
(9, 16419, 69552)
(9, 294627, 1248060)
(9, 5286867, 22395528)
(9, 94868979, 401871444)
(9, 1702354755, 7211290464)
(9, 30547516611, 129401356908)
(9, 548152944243, 2322013133880)
(9, 9836205479763, 41666835052932)
(9, 176503545691491, 747681017818896)
(9, 3167227616967075, 13416591485687196)
(9, 56833593559715859, 240750965724550632)
(9, 1019837456457918387, 4320100791556224180)
(9, 18300240622682815107, 77521063282287484608)

For $m=-3,-2,2,3,6,7,8,10$ no solutions.

Answer 2

Above equation shown below:

$x^2+4xy-y^2=m$ ---$(1)$

"OP' need's many solution's to equation (1).

From a known solution, $(x,y)=(p,q)$ he can get as many

as he likes from the formula which is given below:

$x=(1/2)*(p+q)$

$y=(1/2)*(3p-q)$

It is to be noted that above formula is independent of 'm'

for $m=4$ and $(p,q)=(5,21)$ we get another solution from the

formula above, that is, $(x,y)=(13,-3)$

So instead of computer brute force & instead of trial & error,

from the solution $(13,-3)$ another solution can be arrived at: