STEP II/2018/6

(i) Find all pairs of positive integers (n,p), where p is a prime number, that satisfy

n!+ 5 = p

(ii) In this part of the question you may use the following two theorems:

For n\ge 7, 1! \times 3! \times \cdots \times (2n-1)! > (4n)!\,

For every positive integer n, there is a prime number between 2n and 4n.

Find all pairs of positive integers (n,m) that satisfy

1! \times 3! \times \cdots \times (2n-1)! = m!

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