Shifted primes without large prime factors

Divisors of the Euler and Carmichael functions

“…To state our next result, we define θ to be the supremum of real numbers c so that there are x/log x primes p ≤ x with p − 1 having a prime factor > p c . Many papers have been written on bounding θ, and the current record is θ ≥ 0.677 and is due to Baker and Harman [1]. Theorem 2.…”

Values of the Euler Function in Various Sequences

“…It is known [11] (more recent results can be found in [2] and [18]), that there exists a positive constant δ < 1 such that for all sufficiently large y, the set P = {p ≤ y : p prime and…”

“…According to Theorem 1 from [2], one can take δ = 0.7039. Choosing ε sufficiently small for any given value of k, we obtain the stated result.…”

“…• The problem that has attracted perhaps the most attention, which directly relates the arithmetic properties of λ(n) and n, is the question about the existence of infinitely many Carmichael numbers, that is, composite numbers n for which λ(n) | n − 1 (see [1], as well as the recent improvement given in [2]). …”

On the moments of the Carmichael λ function

“…According to Theorem 1 in [2], one can take δ = .2961. It is conjectured that for each δ > 0, the lower bound (3) holds with any fixed K > 1 for all y > y 0 (δ, K).…”