Nice one, you just wiped some more numbers.
So let's resume. We eliminated :
All odd exponents : 1, 3, 5, 7, 9, ...
All exponents of the form with odd : 2, 6, 10, 14, 18, 22, ...
All exponents of the form : 2, 4, 8, 16, 32, ...
We're still missing a small - yet infinite - subset of numbers, which can be expressed as , , that is, 12, 24, 36, ...
Am I right ?
Here's a start :
Let for some prime number , with .
So :
Therefore, if divides some in , then is divisible by , thus is composite.
So we basically have to show that such a exists for all . I'll give the proof that it doesn't exist for .
Let . Then, hypothetically, for some prime :
Multiply by :
Which is only true for the prime number . However, and we assumed that at the beginning, so we have a contradiction that leads to the conclusion that such a does not exist for .
Anyone can prove that for any exists at least one that satisfies the statement ?
ok so if we have that 10^m+1 is coposite for all odd m, then we assume m is even and so
10^m + 1 = 10^2k +1 = 100^k + 1 and so if k is odd then we have the number is composite, if it is not then we just repeat the process until we have the exponent as an odd number which we will get as m must be a finite, and so there are no more primes of this form, is this helpful??
Best bet at figuring this out is probably modifying Pepin's test and running an optimized computer program:
Pépin's test - Wikipedia, the free encyclopedia