Question

find the number of positive integers n for which 3n-4,4n-5 and 5n-3 all are prime

Answer 1

Answer:

Notice that the sum of the three primes is

(3n−4)+(4n−5)+(5n−3)=12(n−1)

which is clearly even. Therefore, not all of the primes are odd. It follows that one of them has to be equal to the only even prime, 2.

Since n∈N, either 3n−4=2 or 5n−3=2, implying that either n=2 or n=1.

n=1 is rejected since in this case, 3n−4<0. n=2 gives 2,3,7 as our primes. Therefore, n=2 is the only solution to the problem.