|Bill Allombert on Sat, 10 Aug 2013 00:16:47 +0200|
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|Re: x^3 + y^3 = 22 z^3 how to?|
On Fri, Aug 09, 2013 at 03:20:35PM -0600, Elim Qiu wrote: > I'm finding positive int solution(s) (x,y,z) of the equation x^3 + y^3 = 22 > z^3 > other than the multiples of (17299,25469,9954). > > I did some code with iphthon but don't know how to do it with pari > > Could anyone give me some hint please? You are asking for rational point on the curve x^3+y^3 = 22 which is an elliptic curve. By posing u=x+y v=x-y, your equation became u^3+3*v^2*u = 88 Then by posing X = 3*88/u; Y = 9*88*v/u you get Y^2 = X^3-27*88^2 which is a Weierstrass equation, which you can use with ellinit: E=ellinit([0,0,0,0,-27*88]); ? elltors(E) %11 = [1,,] ? ellanalyticrank(E) %6 = [1,4.3180855201574550094703927526599064688] ? P=ellheegner(E) %7 = [553/9,4085/27] So you get a point on E. Using ellpow, you can get others: ? ellpow(E,P,2) %9 = [767848016929/600740100,-672808015029320783/14724139851000] ? ellpow(E,P,3) %10 = [385268181123102953483527273/4808405283271058302221969,5825874533030960746183337375622326929195/10543907772550436016539105119340117703] etc. Now of course, you have to recover the solution of the original equation from X and Y. You get for example: 14176649488890906071282308630139700149971^3+2524900422828984578915633878895046291581^3 =22*5068914033007802762941623871038239169306^3 Cheers, Bill.