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Banks, Ford, Luca, Pappalardi and Shparlinski: values of the Euler function
theorem_4_1: An upper bound of x/(log x)^2 times a subpower of log x for primes p admitting phi(n)=p-1 with n other than p or 2p.
William D. Banks, Kevin Ford, Florian Luca, Francesco Pappalardi and Igor E. Shparlinski, Values of the Euler Function in Various Sequences, Monatshefte für Mathematik 146 (2005), 1--19, DOI:10.1007/s00605-005-0302-7. The publisher records online publication on 6 April 2005.
The edition read is the 25-page author manuscript. Its pagination differs from the journal edition, which has not been compared. The paper studies totient and Carmichael values and totients in polynomial sequences. Its Theorem 4.1 bounds the exceptional shifted-prime values relevant here.
Read status. Claims checked: the exceptional-prime definition on p. 3, the constant on pp. 13--14 and Theorem 4.1 on p. 14 were read from page images. The proof on pp. 14--15 was read for its largest-prime split and totient-value sum, without checking its sieve and counting inputs. No independent proof verification is recorded.
Terms. The author download and publisher record were inspected on 2026-10-15; no open redistribution license was found. No file of this source is held. No full transcription is provided under these terms.
Bears on. Problem 409: the result restricts the primes that can have a nontrivial basin under . It does not settle the exceptional basins.