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

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
2          37,41
3          557, 563, 569
4          1847, 1861, 1867, 1871
(as you can see, I was lazy this week :o)

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
6: 6929381, 6929401, 6929413, 6929423, 6929431, 6929437
7: 134193727, 134193743, 134193757, 134193769, 134193781, 134193793, 134193803

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:

15118087477 15118087529 15118087577 15118087603 15118087627 15118087651 15118087669 15118087679

I have searched up to 40B and not 9 consecutive less than that.

 
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