Timeline for Distribution of fibers of Euler-Phi
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 3, 2012 at 1:47 | comment | added | Gerry Myerson | @Salvo, to be fair to Dickson, that volume was published in 1919. The later "editions" are unaltered reprints of the 1919 text. | |
| Oct 2, 2012 at 10:57 | comment | added | Salvo Tringali | ...a decade before Carmichael himself realized an error in his earlier "proof" (R.D. Carmichael, Note on Euler's $\phi$-Function, Bull. AMS, Vol. 28 (1922), pp. 109-110), with the result that the question is still today broadly open. In spite of the fact that Dickson, in the 2005 edition of his History of the Theory of Numbers (Vol. 1, p. 137), states that the conjecture was proved in Carmichael's original 1907 paper. | |
| Oct 2, 2012 at 10:56 | comment | added | Salvo Tringali | To add something to Felipe's comment, the question is related to a long-standing conjecture, mostly referred to as Carmichael's totient function conjecture, with a story behind it: Carmichael had published an alleged proof (Carmichael, On Euler's $\phi$-Function, Bull. AMS, Vol. 13 (1907), pp. 241-243), and even developed a method of finding a solution for each m (R.D. Carmichael, Notes on the Simplex Theory of Numbers, Bull. AMS, Vol. 15 (1909), pp. 217-223), to the extent of proposing the question as an exercise in his 1914 monograph on the theory of numbers. It took more than... (TBC) | |
| Oct 2, 2012 at 10:23 | comment | added | Salvo Tringali | Yours is the totient valence function, and it is a result of Erdős dating back to 1958 that if for some $m$ there exists at least one $n$ such that $|\phi^{-1}(n)|=m$ then $|\phi^{-1}(n)|=m$ for infinitely many $n$; see P. Erdős, Some Remarks on Euler's $\phi$-Function, Acta Math. (1958), Vol. 4, pp. 10-19. | |
| Oct 2, 2012 at 10:22 | comment | added | Felipe Voloch | Proving that your number is always non-zero is already pretty hard. This reference will get you going: K. Ford, The number of solutions of $\phi(x)=m$ , Annals of Math. 150 (1999), 283--311. | |
| Oct 2, 2012 at 9:25 | history | asked | Jon Cohen | CC BY-SA 3.0 |