0
|
4393 Views
|
3 Replies
|
0 Total Likes
View groups...
Share
GROUPS:

# DIOPHANTINE EQUATION

Posted 9 years ago
 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?
3 Replies
Sort By:
Posted 9 years ago
 HiThere 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
Posted 9 years ago
 HiThere 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
Posted 9 years ago