Compositions with the euler and carmichael functions

Banks, William D.; Luca, Florian; Saidak, Filip; StĂ, P.; nicĂ,

doi:10.1007/bf02942044

Cited by 4 publications

(5 citation statements)

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“…Note that it's not necessary to have any primes on the levels (i, 2). In fact the "worst case scenario" that we will see has no primes on these except Level (1,2). Now that we've described the way to get q a | φ k (n), what is our exponent a?…”

Section: Prime Power Divisors Of φ K (N)mentioning

confidence: 96%

“…We will also turn our attention to finding an asymptotic formula involving iterates involving λ and φ. Banks, Luca, Sȃidȃk, and Stanic in [1] showed that for almost all n, λ(φ(n)) = n exp(−(1 + o(1))(log log n) 2 log log log n) and φ(λ(n)) = n exp(−(1 + o(1))(log log n) log log log n). As a corollary to Theorem 1 we will obtain asymptotic formulas for higher iterates involving λ and φ.…”

Section: Introductionmentioning

confidence: 99%

“…For l ≥ 0 and k ≥ 1, let g(n) = φ l (λ(f (n))), where f (n) is a (k − 1) iterated arithmetic function consisting of iterates of φ and λ. Then the normal order of log(n/g(n)) is 1 (k−1)! (log log n) k log log log n.…”

Section: Introductionmentioning

confidence: 99%

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The iterated Carmichael lambda function

Harland¹

2011

Preprint

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Section: Prime Power Divisors Of φ K (N)mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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The iterated Carmichael lambda function

Harland¹

2011

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“…More advanced techniques for the composition of arithmetic functions are studied in [40], [6], et cetera.…”

Section: (Ii)mentioning

confidence: 99%

Results for Wieferich Primes

Carella

2017

Preprint

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Let v ≥ 2 be a fixed integer, and let x ≥ 1 and z ≥ x be large numbers. The exact asymptotic formula for the number of Wieferich primes p, defined by v p−1 ≡ 1 mod p 2 , in the short interval [x, x + z] is proposed in this note. The search conducted on the last 100 years have produced two primes p < x = 10 15 such that 2 p−1 ≡ 1 mod p 2 . The probabilistic and theoretical information within predicts the existence of another base v = 2 prime on the interval [10 15 , 10 40 ]. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demonstrate that the subset of nonWieferich primes has density 1.

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“…Banks, Luca, Saidak, and Stȃnicȃ [1] studied the the compositions of λ and ϕ. In particular, they studied set of n on which λϕ(n) = ϕλ(n).…”

Section: Propositionmentioning

confidence: 99%