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Getting Primes With 2 And 4 (Posted on 2007-08-19) Difficulty: 2 of 5
(A) Determine all possible prime numbers f such that each of (f+1)/2 and (f-1)/4 are prime numbers.

(B) What are the possible prime numbers g such that each of (g+1)/4 and (g-1)/2 are prime numbers?

  Submitted by K Sengupta    
Rating: 4.0000 (2 votes)
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): f and gPaul2007-08-22 17:55:30
re: f and gK Sengupta2007-08-22 05:13:32
Solutionf and gPaul2007-08-21 15:56:52
Charlie, not Charleyxdog2007-08-20 09:26:41
solution/spoilersxdog2007-08-19 19:31:22
Some Thoughtscomputer exploration -- no proof -- spoilerCharlie2007-08-19 14:44:29
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