Largest known compositorial prime

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# denotesn 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.

SoN 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.

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