Senator 153 wrote:Just out of curiosity... are we thinking that h(100) + 1 is prime? What about "100! + 1"...?
That's a very good question, but it's very difficult to answer. We certainly know that h(100) + 1 is not divisible by any prime less than 50, but it might be divisible by 53, or by 59, or by 97, etc... there's no simple way to prove whether it is or it's not divisible by primes larger than 50. All we could do is calculate the value of h(100) + 1, and attempt long division.
The same is true of 100! + 1; it can't be divisible by primes less than 100, but it might be divisible by 101, or 103, or a lot of other primes. Without doing a superhuman amount of work (superhuman because 100! + 1 is an absolutely enormous number), it is essentially impossible to decide whether 100! + 1 is prime- you'd need a computer. It probably isn't- primes are pretty rare among numbers that large- but there's no way to be sure without getting a computer to work on the problem.
You might think of smaller numbers if you're not convinced- is 5! + 1 prime? We know 5! +1 is not divisible by any prime 5 or less, but that doesn't guarantee 5! + 1 is prime. Indeed, 5! + 1 = 121 = 11^2.; it is not prime. Similarly for 100! + 1; it might be divisible by some prime larger than 100.
