Diophantine equations involving the Euler totient function

Saunders, John C.

doi:10.1016/j.jnt.2019.09.001

Cited by 6 publications

(3 citation statements)

References 8 publications

(6 reference statements)

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“…We thank Daniel Berend for making us aware of the reference [24], and Florian Luca for drawing our attention to some typos in an earlier version of the paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…It is also possible to generalize (2) even further and prove finiteness results for solutions to f (P (x)) = n!, where f is an arithmetic function (see e.g. [24]). For a polynomial P for which it is unknown at present whether (2) has finitely many solutions, such as in the case of the Brocard-Ramanujan problem, one can at least ask for an upper bound on the number of solutions n ≤ N as N → ∞.…”

Section: Introductionmentioning

confidence: 99%

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Power savings for counting solutions to polynomial-factorial equations

Bui¹,

Pratt²,

Zaharescu³

2022

Preprint

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Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n ≤ N to n! = P (x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N ). The proof uses techniques of Diophantine and Padé approximation.2010 Mathematics Subject Classification. 11D45.

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“…We thank Daniel Berend for making us aware of the reference [24], and Florian Luca for drawing our attention to some typos in an earlier version of the paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Power savings for counting solutions to polynomial-factorial equations

Bui¹,

Pratt²,

Zaharescu³

2022

Preprint

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“…Since then, there have been a number of other similar results, in particular, when terms of a Lucas sequence get mapped to terms in the same Lucas sequence, see [1], [3], [5], [6], [7], [12], and [14]. There are also such results and study on the occurrence of the Euler totient function of terms in a Lucas sequence giving a factorial, a power of 2, or a product of a power of 2 and a power of 3, see [4], [9], [15], and [20]. Here we prove the following.…”

Section: Introductionmentioning

confidence: 99%

The Euler Totient Function on Lucas Sequences

Saunders¹

2021

Preprint

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In 2009, Luca and Nicolae [14] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca [7] proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here we generalise these two results and prove that for any fixed natural number P ≥ 3, if we define the sequence (un) n as u0 = 0, u1 = 1, and un = P un−1 + un−2 for all n ≥ 2, then the only solution to the Diophantine equation ϕ (un) = um is ϕ (u1) = ϕ(1) = 1 = u1.

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