For the right hand side in 1), note that the expression is the number of ways to choose n linearly independent vectors from (F_p)^n Since these are the same upto reordering, and there are n! permutations of them, the number of them when u account for reordering (RHS/n!) is still an integer
4 years ago (edited) | 5
I would argue by noting that binomial coefficients are integers. 1) n! divides each product of n consecutive natural numbers and p^(n-1) >= n. 2) As a consequence of my argument for 1), n! divides (n*k+1)*(n*k+2)*...*(n*k+n) for k \in {0, ..., m-1}.
4 years ago | 3
I am attaching a Google drive link for elementary proofs of both the problem 1) uses only Euler's totient theorem , and the expression of exponent of prime in n! 2) simple induction on m https://drive.google.com/file/d/1f7yo9R3Ow4cFX8dYIhQZkZ0HGXsh4HJ9/view?usp=sharing
4 years ago | 0
Mohamed Omar
I came across an interesting phenomenon recently, and that's using algebra techniques to prove interesting number theory facts. For instance, here are couple of examples. In the first one, p is a prime.
These actually come as consequences of using a theorem in group theory called Lagrange's Theorem, and in a nice way.
Do people have thoughts on different ways to prove these instead? Or, do people have ideas on other number theory facts that can be proved in interesting, non-number theoretic ways?
4 years ago (edited) | [YT] | 46