How do we explain the numerical phenomena:
e^(pi*sqrt(163)) ~ 640320^3 + 743.99999999999925...
e^(pi*sqrt(37)) ~ 2^6(6+sqrt(37))^6 - 24.00000138...
e^(pi*sqrt(130)) ~ 12^4(323+40*sqrt(65))^4 - 104.0000000000012...
Are these just flukes or do these numbers fit in a pattern? It turns out the answer is the latter. Quadratic powers sqrt(d) of Gelfond's constant e^pi, numbers which are transcendental, can be closely approximated by certain algebraic numbers. The "excess" of the approximation is also predictable, and approaches various integer values the larger d becomes.
Not to mention that it has an interesting connection to pi formulas and the Monster group...
For number theory, please jump to Page 9 or see this author's "A Collection of Algebraic Identities."
