Divisibility and Congruence problem

[VeeEight]

I was trying to work out whether or not 2n+3 divides (2n+1)! for positive integers n. After trying a few cases I think it does not work but I don't know how a proof for this would work. I tried induction but it got really messy. I also tried rephrasing it, such as putting it into modular equation but have had no luck.

My other question is about a congruence. The statement is that (1-x)^p-1 is congruent to 1 + x + ... + x^p-2 + x^p-1 modulo p where p is an odd prime. I tried to use the binomial theorem to prove this but couldn't finish it because it got really messy and also I have no experience in number theory. Any in understanding and working out these statements would be appreciated.

 

[mathman]

If 2n+3 is prime, it cannot divide (2n+1)! If it is not a prime, it would - factor it into primes and powers of primes. All these factors will be < 2n+1.

 

[VeeEight]

Thanks I actually carried on with my work but thank you for the reply I will investigate it in a minute. And I solved the second question after reading up on fermat's little theorem so no more help is needed. Thank you.

 

[HallsofIvy]

In should have been obvious that 5 does not divide 3!= 6.