necessary MAPLE functions. > read funcs: > readlib(lattice):
> Digits := 30:
> alpha(7,1);> x:=":
> minpoly(x,2);> alpha(7,1/7); > x:=alpha(2,3); > minpoly(x,4); > solve("); > evalf(alpha(2,3)-(3*sqrt(6)-2*sqrt(3))/12,100);
We have used MAPLE's implementation of the lattice algorithm [Bailey-Plouffe].
The MAPLE function minpoly(x,n) finds a polynomial of degree n with small integer coefficients satisfied by the real floating point number x. The output depends on the number of digits computed for the approximation x. In our session the function alpha(p,r) corresponds to . Observe that we were able to verify (3.10) and (3.20) for p=7. We found a nice quartic polynomial which appears to be satisfied by and that it appears thatOnce we have such a conjectured value for it is possible to prove the result by computing to enough digits. In this way we were able to find many initial values. These are given in Table 2.
Table 2: Initial values