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.
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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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Appendix: Computations
Appendix: Computations
Here we provide a PARI/GP implementation of the LR numbers, using a known formula (see, e.g., [4]) along with the recurrence relation (2).
We took about seven months of numerical investigations for primes of the form LR(p,q), p and q odd primes, using the multiprecision software PARI/GP (version 2.3.5) and PFGW (version 3.4.5) through four stages:
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1.
Finding small divisors of the form 2kq±1 with PARI/GP;
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2.
3-PRP tests with PFGW;
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3.
APRCL primality tests for all PRP’s up to 3700 decimal digits with PARI/GP;
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4.
Baillie-PSW PRP tests for all PRP’s above 3700 decimal digits with PARI/GP.
Stage 4 leads to a greater probability of primality than stage 2 (there is no known composite number which is passing this test), but takes more time.
We point out that François Morain provides primality certificates for two large LR numbers (see diamonds ⋄ in Table 3) on his web page. He used his own software fastECPP, implementing a fast algorithm of elliptic curve primality proving [5], on a computer cluster. His calculations required respectively 355 and 2355 days of total CPU time, between January and April 2011. Since LR(157,2207) has 26643 decimal digits, it appears to be the largest prime certification using a general-purpose algorithm, at the date of submission.
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Lygeros, N., Rozier, O. Odd prime values of the Ramanujan tau function. Ramanujan J 32, 269–280 (2013). https://doi.org/10.1007/s11139-012-9420-8
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DOI: https://doi.org/10.1007/s11139-012-9420-8