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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?

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

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Some Thoughts computer exploration -- no proof -- spoiler | Comment 1 of 6

In testing the primes under 100,000,000, only 13 met the first set of criteria, and only 9 and 11 met the second set of criteria.

   10   P=2
   15    loop
   20     T1=(P+1)//2:T2=(P-1)//4
   30     if T1=int(T1) and T2=int(T2) then if prmdiv(T1)=T1 and prmdiv(T2)=T2 then print P,T1,T2
   40     T1=(P+1)//4:T2=(P-1)//2
   50     if T1=int(T1) and T2=int(T2) then if prmdiv(T1)=T1 and prmdiv(T2)=T2 then print tab(40);P,T1,T2
   55     P=nxtprm(P)
   56     if P>100000000 then goto 100
   60    endloop
  100   end

The results were:

                                         3       1       1
 5       3       1
                                         7       2       3
                                         11      3       5
 13      7       3

where the left column reported potential solutions to the first set, and the right column to the second set.  But the first found potential solutions included derived "primes" that were one or both equal to 1, which is non-prime.

  Posted by Charlie on 2007-08-19 14:44:29
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