E^(pi*sqrt(163)) and the solvable sextic 5x^6-640320x^5-10x^3+1 = 0
TPiezas
The sextics,
5x^6-15x^5-10x^3+1 = 0
5x^6-32x^5-10x^3+1 = 0
5x^6-96x^5-10x^3+1 = 0
5x^6-960x^5-10x^3+1 = 0
5x^6-5280x^5-10x^3+1 = 0
5x^6-640320x^5-10x^3+1 = 0
have some very interesting properties.
1. I'm sure some will also recognize the sequence {15, 32, 96, 960,
5280, 640320}. (Hint: Their cubes plus 744 are good approximations to
certain transcendental numbers, with the last as Ramanujan's
constant. The sequence really should be in the OEIS, but it isn't
yet.)
2. Let d = {7,11,19,43,67,163}. With appropriate root "x" of the
sextics chosen respectively, then E^(Pi*Sqrt[d]) ~ (5x)^3 -
6.0000000.... (approx), with more zeroes for the three highest d.
3. The sextics are solvable in radicals, factoring into 2 cubics over
Sqrt(5).
4. The appropriate root is expressible as a Dedekind eta quotient as x
= (1/5)y^2, where y = Exp(Pi*i/6)*DedekindEta[tau] /
DedekindEta[5*tau], with tau = (1+Sqrt[-d])/2. (Note: The symbol "i"
is the imaginary unit.)
More details at http://sites.google.com/site/tpiezas/001
- Titus