Volume 5, 1999, Number 2

R. D. Carmichael [1] conjectured that: “The equation ϕ(x) = A where ϕ is Euler’s totient function, and A is an even positive integer does not have a unique solution. This paper is a continue of paper [5], and it states some theorems and lemmas which help giving an “if and only if” condition for a unique solution, and locate this solution when it exists. By following the same approach of [5] we will be able to determine some particular properties of the set F_A(k+1) , where F_A(k) = {x : ϕ(x) = A_(k)} as defined in [5] is a minimal set of solutions of the equation ϕ(x) = A_(k).

It has been suspected that if Fermat did indeed have a simple proof for his famous ‘last theorem’, that he probably employed his method of infinite descent. In a renewed attempt to see how Fermat might have thought that he had proved that if cⁿ = aⁿ + bⁿ, a, b, c, n ∈ ℤ, n > 2, then a, b, c cannot all be integers, we set c = a + b + m, m ∈ ℤ, and then raised it to the n-th power. The roots of the resulting polynomial in m appear to be – (a + b) only when n ≠ 1 ,2 , and the result might have seemed to Fermat to follow from this. The plausibility of the algebra developed here is considered in the context of the work of the sixteenth century mathematicians, particularly Cardano and Bombelli.