Problems & Puzzles: Puzzles

Puzzle 214.  Binomial coefficient and primes

Sebastián Martín Ruiz obtained and demonstrated the following interesting congruence involving prime numbers and binomial coefficients:

BC(m, pn) = k (mod pn).........(1)

where:

  • BC(x, y) = x!/(y!.(x-y)!)
  • k = ëm/pnű
  • m=>pn
  • pn is the n-th prime number

A corollary of the previous congruence is:

BC(pn+1, pn) = 1 (mod pn)............(2)

because according to the Bertrand's postulate, pn<pn+1<2.pn

Question: Can you demonstrate (1)

Solution:

 
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