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Bulletin of the London Mathematical Society Advance Access originally published online on September 4, 2008
Bulletin of the London Mathematical Society 2008 40(6):1007-1016; doi:10.1112/blms/bdn082
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© 2008 London Mathematical Society

Residue classes having tardy totients

John B. Friedlander

Department of Mathematics
University of Toronto
Toronto
Ontario M5S 2E4
Canada

Florian Luca

Instituto de Matemáticas
Universidad Nacional Autónoma de México
CP 58089, Morelia
Michoacán, México
fluca@matmor.unam.mx

Received 15 September 2007. Revision received 3 June 2008.

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 {phi}(n){equiv} ak (mod mk) exists and that it satisfies log nk/log mk->{infty} as k->{infty}. Here, {phi}(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 {lambda}(n) and the least such n satisfies n<<m82.5.

2000 Mathematics Subject Classification 11B50, 11N64, 11R45.

J.F. was supported in part by NSERC grant A5123 and F.L. was supported in part by grant SEP-CONACyT 46755.


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