Tau Function Prime
المؤلف:
Lehmer, D. H.
المصدر:
"The Primality of Ramanujan,s Tau-Function." Amer. Math. Monthly 72
الجزء والصفحة:
...
22-8-2020
931
Tau Function Prime
is prime for 
, 458329, 942841, 966289, 1510441, ... (OEIS A135430). These values are also known as Lehmer-Ramanujan numbers or LR numbers since the first of them was found by Lehmer (1965). The corresponding primes have explicit values given by 
, 
, ... (OEIS A265913).
It is known that if 
is prime, then 
must be an odd square.
Large values of 
for which 
is a (probable) prime are summarized in the table below (Lifchitz and Lifchitz).
 |
decimal digits |
discoverer |
 |
180524 |
N. Lygeros and O. Rozier (May 2015) |
 |
258571 |
N. Lygeros and O. Rozier (May 2015) |
 |
282048 |
N. Lygeros and O. Rozier (May 2015) |
 |
498503 |
N. Lygeros and O. Rozier (May 2015) |
 |
555339 |
N. Lygeros and O. Rozier (Sep. 2015) |
REFERENCES:
Lehmer, D. H. "The Primality of Ramanujan's Tau-Function." Amer. Math. Monthly 72, 15-18, 1965.
Lifchitz, H. and Lifchitz, R. "PRP Records: Probable Primes Top 10000." https://www.primenumbers.net/prptop/searchform.php?form=Tau%28%3F%29&action=Search.
Sloane, N. J. A. Sequences A135430 and A265913 in "The On-Line Encyclopedia of Integer Sequences."
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