Problems & Puzzles: Puzzles
Puzzle 273. Consecutive 'good' primes
For sure you already know what a good prime is (the name comes from Erdos and Strauss).
pn is a good prime if pn2>pn-i .pn+i , for 1<=i<=n-1
It's also already known that there are infinite of these primes. This was proved by Pomerance according to R. K. Guy (A14, p. 32, UPiNT)
I have calculated the earlier set of k consecutive good primes, for some few values of k.
1.Would you like to extend this table?
2. Can you argue if exist at least one set of k consecutive good primes, for any k value?
Faride Firoozbakht wrote:
5: 216703, 216719, 216731, 216743, 216751
The smallest prime of the earlier set of 8 consecutive good primes is greater than 2145390523.
Jim Fougeron found (August 29, 2004) the earliest set of 8 consecutive good primes: