# Odd prime values of the Ramanujan tau function

@article{Lygeros2013OddPV, title={Odd prime values of the Ramanujan tau function}, author={Nik Lygeros and Olivier Rozier}, journal={The Ramanujan Journal}, year={2013}, volume={32}, pages={269-280} }

We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(pn−1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime.

## 13 Citations

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## References

SHOWING 1-10 OF 14 REFERENCES

### The Primality of Ramanujan's Tau-Function

- Mathematics
- 1965

Although a number of remarkable properties of r(n) have been established, some of which are cited below, there remains a number of unsolved questions about ,r(n); for example: What is the exact order…

### On ℓ-adic representations and congruences for coefficients of modular forms (II)

- Mathematics
- 1977

The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let…

### A proof of Bertrand's postulate

- MathematicsJ. Formaliz. Reason.
- 2012

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate. …

### On Certain Arithmetical Functions Due to

- Mathematics
- 2007

1. Introduction. G. Humbert has discussed, in a series of brief notes, 1 a certain class of entire functions with interesting arithmetical properties. These functions are defined, in an essentially…

### Divisors of Mersenne numbers

- Mathematics
- 1983

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers Mp…

### Existence of Primitive Divisors of Lucas and Lehmer Numbers

- Mathematics
- 2001

We prove that for n > 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

### Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

- Computer Science, MathematicsMath. Comput.
- 2007

The elliptic curve primality proving algorithm is one of the current fastest practical algorithms for proving the primality of large numbers, and an asymptotically fast version, attributed to J. O. Shallit, is described.

### The new book of prime number records

- Mathematics
- 1996

1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.…

### A New Solution to the Equation (p) ≡ 0 (mod p)

- Mathematics
- 2010

The known solutions to the equation �(p) ≡ 0 (mod p) were p = 2, 3, 5, 7, and 2411. Here we present our method to compute the next solution, which is p = 7758337633. There are no other solutions up…

### Divisibilité de certaines fonctions arithmétiques

- Mathematics
- 1975

© Séminaire Delange-Pisot-Poitou. Théorie des nombres (Secrétariat mathématique, Paris), 1974-1975, tous droits réservés. L’accès aux archives de la collection « Séminaire Delange-Pisot-Poitou.…