Odd prime values of the Ramanujan tau function
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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.
KeywordsRamanujan function Primality Lucas sequences
Mathematics Subject Classification (2010)11A41 11F30 11Y11
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.