Carmichael's Totient Function Conjecture - Other Results

Other Results

Ford also proved that if there exists a counterexample to the Conjecture, then a positive fraction (that is infinitely many) of the integers are likewise counterexamples.

Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer n to be a counterexample to the conjecture (Pomerance 1974). According to this condition, n is a counterexample if for every prime p such that p − 1 divides φ(n), p2 divides n. However Pomerance showed that the existence of such an integer is highly improbable. Essentially, one can show that if the first k primes p congruent to 1 (mod q) (where q is a prime) are all less than qk+1, then such an integer will be divisible by every prime and thus cannot exist. In any case, proving that Pomerance's counterexample does not exist is far from proving Carmichael's Conjecture. However if it exists then infinitely many counterexamples exist as asserted by Ford.

Another way of stating Carmichael's conjecture is that, if A(f) denotes the number of positive integers n for which φ(n) = f, then A(f) can never equal 1. Relatedly, Wacław Sierpiński conjectured that every positive integer other than 1 occurs as a value of A(f), a conjecture that was proven in 1999 by Kevin Ford.

Read more about this topic:  Carmichael's Totient Function Conjecture

Famous quotes containing the word results:

    We do not raise our children alone.... Our children are also raised by every peer, institution, and family with which they come in contact. Yet parents today expect to be blamed for whatever results occur with their children, and they expect to do their parenting alone.
    Richard Louv (20th century)

    It is perhaps the principal admirableness of the Gothic schools of architecture, that they receive the results of the labour of inferior minds; and out of fragments full of imperfection ... raise up a stately and unaccusable whole.
    John Ruskin (1819–1900)