--- In

primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

>> Exercise: Find two pairs of positive integers (x,y) such that

4065702994722252685573484796054334194691713593576645739409115721859519

= 5x^2 + 5xy + y^2

Comment: Pari-GP's "qfbsolve" enables a solution in two minutes.

Devotees of "issquare", like Aldrich, may take considerably longer.<<

Here is the 2 minute solution:

R(v)=local(a=G(v[1]),b=G(v[2]));vecsort([X(a*b),X(a*conj(b))],1);

G(p)=qfbsolve(Qfb(5,5,1),p)*[2+u,1]~;

X(z)=vecsort([Z(z),Z(z*u^2),Z(z/u^2)],1)[1];

Z(z)=local(t=z*sign(imag(z)),a=real(t),b=imag(t));[b,max(a-2*b,-a-3*b)];

{A=4065702994722252685573484796054334194691713593576645739409115721859519;

u=quadunit(5);S=R(factorint(A,6)[,1]);T=round(gettime/10^3);

for(k=1,2,print("x="S[k][1]);print("y="S[k][2]));print(T" seconds");}

x=8742252927800705984069016007531927

y=42652016277694077218687373764822402

x=24507765935843939778835994638318507

y=8131530641557000519732930155331538

119 seconds

David