Part A
Since f is prime, it follows that f = 2, 3 or f must possess the form 6n + 1 or 6n-1 for positive integers n.
(f-1)/4 is prime, and so, (f-1)/4 >= 2, so that f >=9. Hence, f cannot be equal to 2 or 3.
Accordingly f = 6n+1, or 6n-1
In the former case, (f-1)/4 = 3n/2, and so n must be even, but for even n> 2, we observe that (f-1)/4 is composite, and accordingly, n= 2. This gives, f = 13, so that (f+1)/2 =7; which is a prime number.
In the latter case, (f+1)/2 = 3n which is composite for n> 1, and so n=1, which gives f = 5, and so (f-1)/4 = 1, which is not prime and thus leads to a contradiction.
Consequently, f = 13 is the only possible solution.
Part B
Since g is prime, it follows that g = 2, 3 or g must possess the form 6n + 1 or 6n-1, for positive integers n.
(g+1)/4 is prime, and so, (g +1)/4 >= 2, so that f >=7. Hence, g cannot be equal to 2 or 3.
Accordingly g = 6n+1, or 6n-1
If g = 6n+1, then (g-1)/2 = 3n, which is prime iff n = 1, and (g+1)/4 = (3n+1)/2 = (3*1+1)/2 = 2
Hence g = 6*1 + 1 = 7
If g = 6n-1, then ((g+1)/4 = 3n/2 , which is prime iff n = 2, and (g-1)/2 = 3n-1 = (3*2-1)= 5
Hence g = 6*2 - 1 = 11
Consequently, g = 7, 11 are the only possible solutions.
|
