(5 * 10[sup]25[/sup])[sup](5 * 10[sup]25[/sup])[/sup]??

When it's about 10↑(10↑25) digits?? Woah. Nothing until then. For now, let's consider 5 the smallest value for which this is true.

[QUOTE=3.14159;230187](5 * 10[sup]25[/sup])[sup](5 * 10[sup]25[/sup])[/sup]??

When it's about 10↑(10↑25) digits?? Woah. Nothing until then. For now, let's consider 5 the smallest value for which this is true.[/QUOTE]

Not that bad -- (5 * 10[sup]25[/sup])[sup]5 * 10[sup]25[/sup][/sup] has 'only' about 10[sup]27[/sup] digits. But it's certainly far beyond the range that could be tested (or even stored!).

[QUOTE=Charles]But it's certainly far beyond the range that could be tested (or even stored!).[/QUOTE]

If the counterexample is so large that it cannot be tested, let's simply assume 5 is the smallest.

what is the probability that there is a prime below 1,000,000 (for example) digits?

[QUOTE=Dougal]what is the probability that there is a prime below 1,000,000 (for example) digits?
[/QUOTE]

1 in 2302586.

[QUOTE=3.14159;230200]1 in 2302586.[/QUOTE]

that means we'd need to check above on average 10^999993-10^999994

An example of a prime below 10[sup]6[/sup] digits is 912646 * 798336[sup]20160[/sup] + 1 (118995 digits).

And; to be random; I think someone posted here that there was a 337-digit composite discovered that was pseudoprime to all primes below 200.

what is the probability that there is a prime of the form 5*b^b+1 below 1,000,000 (for example) digits?

[QUOTE=Dougal]what is the probability that there is a prime of the form 5*b^b+1 below 1,000,000 (for example) digits?
[/QUOTE]

See post 176. There should be none smaller than 10[sup]27[/sup] digits.

[QUOTE=3.14159;230216]See post 176. There should be none smaller than 10[sup]27[/sup] digits.[/QUOTE]

no.we'd expect [B]ONE[/B] below that,what is the chance,that,that one is below 1,000,000 digits?