Lower Bounds
There are very high lower bounds for Carmichael's conjecture that are relatively easy to determine. Carmichael himself proved that any counterexample to his conjecture (that is, a value n such that φ(n) is different from the totients of all other numbers) must be at least 1037, and Victor Klee extended this result to 10400. A lower bound of was given by Schlafly and Wagon, and a lower bound of was determined by Kevin Ford in 1998.
The computational technique underlying these lower bounds depends on some key results of Klee that make it possible to show that the smallest counterexample must be divisible by squares of the primes dividing its totient value. Klee's results imply that 8 and Fermat primes (primes of the form 2k+1) excluding 3 do not divide the smallest counterexample. Consequently, proving the conjecture is equivalent to proving that the conjecture holds for all integers congruent to 4 (mod 8).
Read more about this topic: Carmichael's Totient Function Conjecture
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—Samuel Beckett (1906–1989)
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