Carmichael's conjecture states that if φ (x) = n, then φ (y) = n for some y ≠ x (φ is Euler's totient function). We show that the conjecture is valid for all x under 10 10,900,000. The main new idea ...

Second, the general applicability of Euler's Theorem results to nonhomogeneous (variable-proportional-return) cases is developed. Lastly, it is shown that short-run, nonhomogeneous production/yield ...

A stretch too far, perhaps, but Nahin’s tale of the formula ei π + 1 = 0, which links five of the most important numbers in mathematics, is remarkable.