Wikipedia:Reference desk/Archives/Mathematics/2022 November 15

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November 15

Largest prime with a known index

While the largest known prime number is 2^82,589,933 − 1, its exact index in the sequence of all prime numbers is not known. What is the largest prime number for which its exact index in the sequence of all prime numbers is known? GeoffreyT2000 (talk) 16:20, 15 November 2022 (UTC)

It's not a simple question, it seems, but this Reddit thread from 2016 discusses it and may lead you to some interesting places. This Quora thread from the same time period also discusses the matter, and seems to have some better evidence for a prime that meets your definition. The second thread cites work by Tomás Oliveira e Silva that has set a lower bound for such a sequence of primes somewhere on the order of 4 x 10¹⁸. --Jayron32 17:11, 15 November 2022 (UTC)

Those sources are about the largest prime for which all smaller primes have been identified. The question only asked for the index, meaning we only have to know the number of smaller primes. There are faster ways to compute this number than to compute and count each of the primes. See Prime-counting function. The current record is pi(10²⁹) = 1,520,698,109,714,272,166,094,258,063.[1] To make it an index of a prime, compute the largest prime below 10²⁹ which is 10²⁹−27. You could technically improve the record slightly by computing and counting the next few primes above 10²⁹ but that is not considered an official record. Any such "record" could keep being broken in a small fraction of a second if you assume the previous count was correct. PrimeHunter (talk) 19:22, 15 November 2022 (UTC)

Well, there you go. Thanks for the clarification. --Jayron32 19:33, 15 November 2022 (UTC)