On a Diophantine equation involving Fibonacci numbers and the Ramanujan $$\tau $$-function of factorials

Luca, Florian; Mabaso, Sibusiso

doi:10.1007/s13370-021-00950-1

Cited by 3 publications

(6 citation statements)

References 13 publications

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“…• d (m) = 1. Then t = 2 i for some i ≥ 0, and by (10) we have a = (2 i − 3) /5. By Lemma 2.2, i ≡ 3(mod 4).…”

Section: N Is Not Square-free With N Oddmentioning

confidence: 99%

“…It is clear from (10) that d (m) is odd and cannot be ≥ 6. Now, we consider separately the following possibilities:…”

Section: N Is Not Square-free With N Oddmentioning

confidence: 99%

“…(2 y q t 1 +1)(2 z q t ′ 1 +1), where q 1 is a Fermat prime and y, z, t, t ′ are positive integers, with (q 1 , y + z) ∈ { (3,11), (5,10), (17, 8), (257, 4)} and 2 y q t 1 + 1, 2 z q t ′ 1 + 1 are primes.…”

mentioning

confidence: 99%

“…2 y q t 1 + 1 2 z q t ′ 1 + 1 , where 2 x + 1, 2 y q t 1 + 1 and 2 z q t ′ 1 + 1 are primes with 2 y q t 1 < 2 z q t ′ 1 and 5 | 13(t + t ′ ) − 9, and we have (q 1 , y + z) ∈ { (3,11), (5,10), (17, 8), (257, 4)}. For example, for x = t = 1, y = 5, z = 6 and t ′ = 2 we get n = 3 6 • 97 • 577.…”

mentioning

confidence: 99%

“…• q 1 q 17t−9 2 (2 z q t 2 + 1), where q 1 , q 2 are Fermat primes and z, t are positive integers with (q 1 , q 2 , z) ∈ {(3, 5, 13), (3,17,11), (3,257,7), (5,17,10), (5,257,6), (17, 257, 4)} and 2 z q t 2 + 1 is prime.…”

mentioning

confidence: 99%

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A class of solutions of the equation d(n2) = d(φ(n))

Amroune,

Bellaouar,

Boudaoud

2023

NNTDM

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For any positive integer $n$ let $d\left( n\right) $ and $\varphi \left( n\right) $ be the number of divisors of $n$ and the Euler's phi function of $n$, respectively. In this paper we present some notes on the equation $d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right).$ In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson's conjecture implies that $d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right) $ infinitely often.

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“…• d (m) = 1. Then t = 2 i for some i ≥ 0, and by (10) we have a = (2 i − 3) /5. By Lemma 2.2, i ≡ 3(mod 4).…”

Section: N Is Not Square-free With N Oddmentioning

confidence: 99%

“…It is clear from (10) that d (m) is odd and cannot be ≥ 6. Now, we consider separately the following possibilities:…”

Section: N Is Not Square-free With N Oddmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%