Euler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another.

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 ...

THOMAS J. OSLER, 92.43 Euler's little summation formula and special values of the zeta function, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 295-299 ...