13 x^2+1=y^2, what is the procedure to solve the equation with integer solutions, wolfram gives the infinite integer solution for the equation, but how do they get it?
Hi
There are various ways to ‘solve’ that equation, If we just asked to solve it like
Clear[x, y]; Solve[13 x^2 + 1 == y^2, {x, y}, Integers]
We will get results like
{{x -> ConditionalExpression[-(((649 - 180 Sqrt[13])^
C[1] - (649 + 180 Sqrt[13])^C[1])/(2 Sqrt[13])),
C[1] \[Element] Integers && C[1] >= 0],
which isn’t very good, this is telling me that there could be an infinite number of solutions and that we need to add limits for x and y, so we could ask this
Clear[x, y]; Solve[
13 x^2 + 1 == y^2 && 0 < x < 10^10 && 0 < y < 10^10, {x,
y}, Integers]
{{x -> 180, y -> 649}, {x -> 233640, y -> 842401}, {x -> 303264540,
y -> 1093435849}}
We could also ask this
Clear[x, y]; FindInstance[13 x^2 + 1 == y^2, {x, y}, Integers, 2]
{{x -> -98475707056574514234147687427309858128133932917550159192926078\
9721440877383880976428933444437557685123880,
y -> -35505921118005041063603236141349722598219803651103555766541663\
27589596496957941863373110484725700882173601}, {x ->
1187828648027618759642502802027120177356001334838507471515621279589\
8286867954206065456648483958433444781041798143695420315713621488215926\
4598844841133760412740750545423228379788307197484815120271546811447749\
7159652848682155570439974210170748217810526303751922055046373538138102\
0231971292940018133726241543544132192067186548069882096119540954036932\
577807293981659210167396951286006140,
y -> 428277709692864682878803963175563587116293964276476908497499183\
3145609033134526228869899336669731155196749772066652590596524997159085\
2519889096595069946728890727914390171744935742586153090418212768714274\
7310625391649304220246973177349306393132073180127067871356808850048354\
1013054379064542178999843193995372857640443443929933173325176356363542\
2922385906493065904876438405625228363849}}
Hope that helps a little
Hi
There are various ways to ‘solve’ that equation, If we just asked to solve it like
Clear[x, y]; Solve[13 x^2 + 1 == y^2, {x, y}, Integers]
We will get results like
{{x -> ConditionalExpression[-(((649 - 180 Sqrt[13])^
C[1] - (649 + 180 Sqrt[13])^C[1])/(2 Sqrt[13])),
C[1] \[Element] Integers && C[1] >= 0],
which isn’t very good, this is telling me that there could be an infinite number of solutions and that we need to add limits for x and y, so we could ask this
Clear[x, y]; Solve[
13 x^2 + 1 == y^2 && 0 < x < 10^10 && 0 < y < 10^10, {x,
y}, Integers]
{{x -> 180, y -> 649}, {x -> 233640, y -> 842401}, {x -> 303264540,
y -> 1093435849}}
We could also ask this
Clear[x, y]; FindInstance[13 x^2 + 1 == y^2, {x, y}, Integers, 2]
{{x -> -98475707056574514234147687427309858128133932917550159192926078\
9721440877383880976428933444437557685123880,
y -> -35505921118005041063603236141349722598219803651103555766541663\
27589596496957941863373110484725700882173601}, {x ->
1187828648027618759642502802027120177356001334838507471515621279589\
8286867954206065456648483958433444781041798143695420315713621488215926\
4598844841133760412740750545423228379788307197484815120271546811447749\
7159652848682155570439974210170748217810526303751922055046373538138102\
0231971292940018133726241543544132192067186548069882096119540954036932\
577807293981659210167396951286006140,
y -> 428277709692864682878803963175563587116293964276476908497499183\
3145609033134526228869899336669731155196749772066652590596524997159085\
2519889096595069946728890727914390171744935742586153090418212768714274\
7310625391649304220246973177349306393132073180127067871356808850048354\
1013054379064542178999843193995372857640443443929933173325176356363542\
2922385906493065904876438405625228363849}}
Hope that helps a little
