I ran across a blog post here that said a new record has been set for the largest compositorial prime. [1]
OK, so what is a compositorial prime? It is a prime number of the form
n! / n# + 1
where n# denotes n primorial, the product of the prime numbers no greater than n.
The newly discovered prime is
N = 751882!/751882# + 1
It was described in the article cited above as
751882!/751879# + 1,
but the two numbers are the same because there are no primes greater than 751879 and less than 751882, i.e.
751879# = 751882#.
About how large is N? We can calculate the log of the numerator easily enough:
>>> import scipy.special >>> scipy.special.loggamma(751883) 9421340.780760147
However, the denominator is harder to compute. According to OIES we have
n# = exp((1 + o(1)) n)
which would give us the estimate
log(751882#) ≈ 751882.
So
log10(N) = log(N) / log(10) ≈ (9421340 − 751882) / log(10) ≈ 3765097.
According to this page,
log10(N) = 3765620.3395779
and so our approximation above was good to four figures.
So N has between 3 and 4 million digits, making it much smaller than the largest known prime, which has roughly 41 million digits. Overall, N is the 110th largest known prime.
[1] I misread the post at first and thought it said there was a new prime record (skipping over the “compositorial” part) and was surprised because the number is not a Mersenne number. For a long time now the largest known prime has been a Mersenne prime because there is a special algorithm for testing whether Mersenne numbers are prime, one that is much more efficient than testing numbers in general.