Bill Allombert on Sun, 11 Aug 2013 22:39:08 +0200 |
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Re: x^3 + y^3 = 22 z^3 how to? |
On Sat, Aug 10, 2013 at 12:16:34AM +0200, Bill Allombert wrote: > 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]); I meant E=ellinit([0,0,0,0,-27*88^2]); obviously. The remainder is correct notwithstanding. > ? 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] Cheers, Bill.