# Odd prime values of the Ramanujan tau function

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## Abstract

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*(*p* ^{ n−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.

## Keywords

Ramanujan function Primality Lucas sequences## Mathematics Subject Classification (2010)

11A41 11F30 11Y11## Notes

### Acknowledgements

We are grateful to François Morain for his outstanding contribution to the numerical results. We also would like to thank Marc Hufschmitt and Paul Zimmermann for their helpful discussions, and the anonymous referee for his valuable suggestions.

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