Is this number prime?

[SLD]

(10^200!)!+1

Obviously it's way beyond the capacity to calculate or even express this number in scientific notation. but can you determine whether it is prime?

The largest prime is 2^82,589,933 − 1.

if not, can you determine the probability that it is prime? I would think it to be pretty low.

Or how far away the closest prime should be?

 

[Jimmy Higgins]

Is there a rate of declination of numbers being prime the further down the number line you go?

 

[SLD]

Is there a rate of declination of numbers being prime the further down the number line you go?

Certainly. Just look at the list of the first 100 primes and you can see that trend.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541

I also checked primes in the 56 million range. just randomly picked 56,432,101 to 56,432,199 and there are only two, 56,432,107 and 56,432,191. There are 25 primes from 2 to 100.

ETA: I don’t here’s a specific formula that shows how much they decline though.

 

[Jimmy Higgins]

The rate declination would help to provide a statistical chance.

 

[lpetrich]

ETA: I don’t here’s a specific formula that shows how much they decline though.

There is:  Prime number theorem


For P(n) = number of primes <= n
P(n) -> n/log(n)
where log is the natural logarithm


 Prime number mentions some primality tests, and  List of prime numbers has some lists of various kinds of primes.

Among them are  Factorial prime primes like the OP's prime.

A002982 - OEIS -- Numbers n such that n! - 1 is prime.
3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003

A002981 - OEIS -- Numbers k such that k! + 1 is prime.
0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465

(From that entry) Comments: If n + 1 is prime then (by Wilson's theorem) n + 1 divides n! + 1. Thus for n > 2 if n + 1 is prime n is not in the sequence. - Farideh Firoozbakht, Aug 22 2003

Returning to the OP's prime, I used Wilson's theorem and checked on whether 10^200 + 1 is a prime. It isn't, so that criterion does not give us anything further. It shows us that (10^200)! + 1 is not divisible by 10^200 + 1, but it does not tell us what other nontrivial divisors it might have, if any.

Alhacen-Lagrange-Theorem --  Wilson's theorem