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). p_{n} is a good prime if p_{n}^{2}>p_{ni} .p_{n+}_{i} , for 1<=i<=n1 See A028388. 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. K
Primes Questions: 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? Solution: Faride Firoozbakht wrote: k: Primes 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:





