00:01
In this question, we want to find five integers of the form 2 to the power of n minus 1 that are composite if n is prime.
00:12
And really, this is going to be a giant guess and check operation.
00:17
But if you do, i'll show you five such integers.
00:22
So, for example, the first n value that's prime, where 2 to the n minus 1 is composite, turns out to be 11.
00:33
So, if n is 11, then 2 to the power of n minus 1, if you go to a calculator, 2 to the 11th power minus 1 ends up being 2047.
00:51
And the question is, what is this factorization? well, if you go to a calculator, you'll be able to find that indeed this factors as 23 times 89.
01:05
So, n is prime, but 2 to the n minus 1 is composite.
01:11
The next value of n that's prime, where 2 to the n minus 1 is composite, is 23.
01:18
In this case, 2 to the power of n minus 1 is 8 ,388 ,607.
01:32
And its factorization ends up being 47 times 178 ,481.
01:43
The next value of n that's prime, leading to 2 to the n minus 1 being composite, is 29.
01:52
2 to the power of 29 minus 1 is going to be a gigantic number.
01:57
And i'm just copying this down.
01:59
It's going to be 0 ,911.
02:05
So, it's going to be 536 ,870 ,911...
