In mathematics, **Carmichael's totient function conjecture** concerns the multiplicity of values of Euler's totient function φ(*n*), which counts the number of integers less than and coprime to *n*. It states that, for every *n* there is at least one other integer *m* ≠ *n* such that φ(*m*) = φ(*n*). Robert Carmichael first stated this conjecture 1907, but as a theorem rather than as a conjecture. However, his proof was faulty and in 1922 he retracted his claim and stated the conjecture as an open problem.

Read more about Carmichael's Totient Function Conjecture: Examples, Lower Bounds, Other Results

### Famous quotes containing the words conjecture, function and/or carmichael:

“What these perplexities of my uncle Toby were,—’tis impossible for you to guess;Mif you could,—I should blush ... as an author; inasmuch as I set no small store by myself upon this very account, that my reader has never yet been able to guess at any thing. And ... if I thought you was able to form the least ... *conjecture* to yourself, of what was to come in the next page,—I would tear it out of my book.”

—Laurence Sterne (1713–1768)

“The *function* of the actor is to make the audience imagine for the moment that real things are happening to real people.”

—George Bernard Shaw (1856–1950)

“An impersonal and scientific knowledge of the structure of our bodies is the surest safeguard against prurient curiosity and lascivious gloating.”

—Marie *Carmichael* Stopes (1880–1958)