Let p be a prime number greater than 2. Prove that 2/ p can be expressed in exactly one way in the form 1 /m + 1 /n where m and n are positive integers with n > m

Answer 1

1/m + 1/n = (n+m)/mn

(n+m)/mn = 2/p

Thus, p|mn

As n > m, 1/m > 1/p, so m < p, and p|n

m | n, or the denominator of 1/m + 1/n must contain a factor other than p, which would be a contradiction

Thus, 1/m + 1/cpm = 2/p for some c.

(cp+1)/cpm = 2/p

(cp+1)/cm = 2

2cm = cp + 1

Thus, as c | 2cm and c|cp, c|1, so c = 1

Thus, 1/m + 1/pm = 2/p

(p+1)/pm = 2/p

2m = p+1

m = (p+1)/2

n = p(p+1)/2