(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?

(A) Determine all possible prime numbers f
such that each of (f+1)/2 and (f-1)/4 are prime numbers.

To ease reference let g=(f+1)/2 and h=(f-1)/4.

A1--For f=2, neither g nor h is integral.

A2--If f is of the form 4k+1, g=2k+1 and h=k. 

A2a-For k=2, f is non-prime. 

A2b-For k=3, f and g are both prime, giving Charley's solution f=13.

A2c-For k>3, if k=6j+1 then g=12j+3 non-prime, and if k=6j-1 then f=24j-3 non-prime.

A3-If f is of the form 4k+3, g=2k+2 non-prime for k>0.

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

Let h=(g+1)/4 and i=(g-1)/2.

B1--For g=2, neither h nor i is integral.

B2--If g is of the form 4k+1, h is non-integral for k>1. 

B3-For g=4k+3, h=k+1 and i=2k+1. 

B3a-For k=1, g=7, h=2, i=3, found by Charley.

B3b-For k=2, g=11, h=3, i=5, also found by Charley.

B3c-For h>3 k is even. 

B3d-If k=6j then g=24k+3 non-prime; if k=6j+2 then h=6j+3 non-prime; if k=6j+4 then i=12j+9 non-prime.

Posted by xdog on 2007-08-19 19:31:22