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Tau Function Prime

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A tau function prime is a prime number of the form |tau(n)|, where n is a positive integer and tau(n) is the tau function. The values of n for which |tau(n)| is an odd prime number begin with 63001, 458329, 942841, 966289, 1510441, ... (OEIS A135430). The first was found by Lehmer (1965). The corresponding signed values of the tau function begin with -80561663527802406257321747, -11695495424911987900947041440697, ... (OEIS A265913).

Lygeros and Rozier (2013) define Lehmer-Ramanujan numbers or LR numbers by LR(p,q)=tau(p^(q-1)) for odd primes p and q.

If |tau(n)| is an odd prime number, then n is an odd square (Lygeros and Rozier 2013).

Large values of n for which |tau(n)| was reported to be a prime number or probable prime are summarized in the table below (Lifchitz and Lifchitz).

tau(n)decimal digitsdiscoverer
tau(643^(11688))180524N. Lygeros and O. Rozier (May 2015)
tau(971^(15738))258571N. Lygeros and O. Rozier (May 2015)
tau(509^(18946))282048N. Lygeros and O. Rozier (May 2015)
tau(631^(32370))498503N. Lygeros and O. Rozier (May 2015)
tau(773^(34960))555339N. Lygeros and O. Rozier (Sep. 2015)

See also

Integer Sequence Primes, Tau Function

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References

Lehmer, D. H. "The Primality of Ramanujan's Tau-Function." Amer. Math. Monthly 72, 15-18, 1965.Lifchitz, H. and Lifchitz, R. "PRP Top Records: Search for Tau(?)." http://www.primenumbers.net/prptop/searchform.php?form=Tau%28%3F%29&action=Search.Lygeros, N. and 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.Sloane, N. J. A. Sequences A135430 and A265913 in "The On-Line Encyclopedia of Integer Sequences."

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Tau Function Prime

Cite this as:

Weisstein, Eric W. "Tau Function Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TauFunctionPrime.html

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