Problem:
For each positive integer m>1, let P(m) denote the greatest prime factor of m. For how many positive integers n is it true that both P(n)=n​ and P(n+48)=n+48​?
Answer Choices:
A. 0
B. 1
C. 3
D. 4
E. 5
Solution:
The conditions imply that both n and n+48 are squares of primes. So for each successful value of n we have primes p and q with p2=n+48 and q2=n, and
48=p2−q2=(p+q)(p−q)
The pairs of factors of 48 are
48 and 1,24 and 2,16 and 3,12 and 4, and 8 and 6.
These give pairs (p,q), respectively, of
(249​,247​),(13,11),(219​,213​),(8,4), and (7,1)
Only (p,q)=(13,11) gives prime values for p and for q, with n=112=121 and n+48=132=169.