Problem: For n≥50 the number of prime numbers greater than n!+1 and less than n!+n, is: [n!=1⋅2⋯(n−1)⋅n; thus: 3!=1⋅2⋅3=6;5!=1⋅2⋅3⋅4⋅5=120]
Answer Choices:
A. 0
B. 1
C. n−1
D. n
E. 2n for n even, 2n+1 for n odd
Solution:
Consider the integer n!+k where 1<k<n. Since n ! contains each of the factors 1,2 , 3,…,n, it contains the factor k. Since n!+k can be written as the product of two factors, one of which is k, it is composite. Hence, there are no primes between n!+1 and n!+n.
Comment. The conclusion holds for n≧1.