Which number is larger?

Factual Questions
1

Which number is larger:

3,756,801,695,685 x 2 ^666,669 ^=to the power of
(the largest known prime)

OR

(10^87 -1)! ^=to the power of
(The number of connections among all the particles in the observable universe)

If it is the case that the number of connections connecting every particle to every other particle is far greater than the largest currently-known prime, there are nonetheless an infinite number of primes larger than it (because the primes are infinite and the number of connections is finite).

2

The first number is not prime, and I don’t know what you mean by connections, but the second number is a lot bigger than the first number.

3

It looks like the OP meant the exponent of the first number to be the largest known prime, in which case he forgot the -1 (largest known primes are always of the form (2^n -1) ). But that’s an out of date value; the current largest known prime is 2^77,232,917 − 1

4

Well, in any case,

2[sup]77232917[/sup]-1 ≈ 5 × 10[sup]23249424[/sup]

and (10[sup]87[/sup] -1)! ≈ 10[sup]10[sup]89[/sup][/sup]

5

The largest prime needs to be multiplied by about 3.75 trillion, but that only adds 13 to the exponent. Won’t make any difference to the comparison.