Residue classes having tardy totients

Abstract

It is shown, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n=nk of the congruence ϕ(n)≡ ak (mod mk) exists and that it satisfies log nk/log mk→∞ as k→∞. Here, ϕ(n) is the Euler function. This answers a question raised by the first author and Shparlinski (Bull. London Math. Soc. 39 (2007) 425–432; Bull. London Math. Soc. 40 (2008) 532). It is also shown that every congruence class ak (mod mk) containing an even integer contains infinitely many values of the Carmichael function λ(n) and the least such n satisfies n≪m82.5.

Locations

• Bulletin of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View