Discover an identity using lattice basis reduction

Find integers a and b such that a*Zeta(50)+b*Pi^50 = 0 using lattice basis reduction .

First try using quadruple precision.

> readlib(lattice):

> Digits := 32;

[Maple Math]

> a1 := Zeta(50.);

[Maple Math]

> a2 := evalf(Pi^50);

[Maple Math]

> LB := [[1,0,round(a1*10^(Digits-2))],[0,1,round(a2*10^(Digits-2))]];

[Maple Math]
[Maple Math]

> reduced_LB := lattice(LB);

[Maple Math]
[Maple Math]
[Maple Math]

> a := reduced_LB[1][1];

[Maple Math]

> b := reduced_LB[1][2];

[Maple Math]

Now redo the problem using 80 digit precision to see the correct answer.

> Digits := 80;

[Maple Math]

> a1 := Zeta(50.);

[Maple Math]
[Maple Math]

> a2 := evalf(Pi^50);

[Maple Math]
[Maple Math]

> LB := [[1,0,round(a1*10^(Digits-2))],[0,1,round(a2*10^(Digits-2))]];

[Maple Math]
[Maple Math]
[Maple Math]
[Maple Math]

> reduced_LB := lattice(LB);

[Maple Math]
[Maple Math]
[Maple Math]
[Maple Math]
[Maple Math]
[Maple Math]

> a := reduced_LB[1][1];

[Maple Math]

> b := reduced_LB[1][2];

[Maple Math]

Check by using higher precision, also by using [Maple Math] .

> Digits := 2*Digits;

[Maple Math]

> almost_zero_2 := a*Zeta(50.)+b*evalf(Pi^50);

[Maple Math]

> z50 := expand(Zeta(50));

[Maple Math]

> z50r := op(1,z50);

[Maple Math]

> zero_2 := z50r+b/a;

[Maple Math]