On the prime factors of the iterates of the Ramanujan <i>τ</i>–function

Luca, Florian; Mabaso, Sibusiso; Stănică, Pantelimon

doi:10.1017/s0013091520000310

Cited by 2 publications

(4 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We should also comment on the particular choice of the constant "13" on the right hand side of inequality (9) in Corollary 1.1 (and analogously in Theorem 5). As we shall observe, the weaker result with 13 replaced by 11 (corresponding to equation (10) with δ = 0) reduces via local arguments to the resolution of a single Thue equation; this is the content of Proposition 6 of Luca et al [24]. Corollary 1.1 as stated requires (apparently at least) the full use of our various techniques, including the Primitive Divisor Theorem, solution of a variety of Thue-Mahler equations, and resolution of hyperelliptic equations through appeal to the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves.…”

Section: Theoremmentioning

confidence: 77%

Odd values of the Ramanujan tau function

et al. 2021

View full text Add to dashboard Cite

We prove a number of results regarding odd values of the Ramanujan τ -function. For example, we prove the existence of an effectively computable positive constant κ such that if τ (n) is odd and n ≥ 25 then eitherlog log log n log log log log n or there exists a prime p | n with τ ( p) = 0. Here P(m) denotes the largest prime factor of m. We also solve the equation τ (n) = ±3 b 1 5 b 2 7 b 3 11 b 4 and the equations τ (n) = ±q b where 3 ≤ q < 100 is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue-Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.Communicated by Kannan Soundararajan. Michael A. Bennett is supported by NSERC. Adela Gherga and Samir Siksek are supported by an EPSRC Grant EP/S031537/1 "Moduli of elliptic curves and classical Diophantine problems".

show abstract

Section: Theoremmentioning

confidence: 77%

Odd values of the Ramanujan tau function

et al. 2021

View full text Add to dashboard Cite

show abstract

“…We should also comment on the particular choice of the constant "13" on the right hand side of inequality (9) in Corollary 1.1 (and analogously in Theorem 5). As we shall observe, the weaker result with 13 replaced by 11 (corresponding to equation (10) with δ = 0) reduces via local arguments to the resolution of a single Thue equation; this is the content of Proposition 6 of Luca, Maboso, Stanica [23]. Corollary 1.1 as stated requires (apparently at least) the full use of our various techniques, including the Primitive Divisor Theorem, solution of a variety of Thue-Mahler equations, and resolution of hyperelliptic equations through appeal to the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves.…”

Section: Introductionmentioning

confidence: 83%

“…For future use, it will be of value for us to record some basic arithmetic facts about τ (n); these are taken from Swinnerton-Dyer's article [37]. Here σ v (n) denotes the sum of the v-th powers of the divisors of n. 23) for other p = 23 (16) τ (n) ≡ σ 11 (n) (mod 691).…”

Section: Congruences For the τ Functionmentioning

confidence: 99%

See 1 more Smart Citation

Odd values of the Ramanujan tau function

Bennett¹,

Gherga²,

Patel³

et al. 2021

Preprint

View full text Add to dashboard Cite

We prove a number of results regarding odd values of the Ramanujan τ -function. For example, we prove the existence of an effectively computable positive constant κ such that if τ (n) is odd and n ≥ 25 then eitherlog log log n log log log log n or there exists a prime p | n with τ (p) = 0. Here P (m) denotes the largest prime factor of m. We also solve the equation τ (n) = ±3 b 1 5 b 2 7 b 3 11 b 4 and the equations τ (n) = ±q b where 3 ≤ q < 100 is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue-Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

show abstract

On the prime factors of the iterates of the Ramanujan τ–function

Abstract: In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

Cited by 2 publications

References 14 publications

Odd values of the Ramanujan tau function

Odd values of the Ramanujan tau function

Odd values of the Ramanujan tau function

Contact Info

Product

Resources

About