Problems & Puzzles: Puzzles

Puzzle 58.- A particular sequence of k consecutive primes (by Enoch Haga)

Let's consider the pair of primes (p, q) such that q = 4p² + 1 (See Puzzle 54)

Let's ask for k consecutive primes p of this kind.

It's easy to show that the earliest k primes for k = 4 & 5 are:

k=4: p = 2, 3, 5 & 7 (C. Rivera)
k=5: p = 1627, 1637, 1657, 1663 & 1667 (C. Rivera)

Find the earliest k = 6, 7 & 8 consecutive p primes of this type. 

Felice Russo wrote at 11/06/99 "this is to inform you that I didn't find any other solution up to p=<10185944149".

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Giovanni Resta wrote (Nov. 2004):

Contrary to what Russo wrote, there exists a solution for
k=6 in the range he searched. It is:

1648002773, 1648002787, 1648002793, 1648002803, 1648002823, 1648002827 and the corresponding 4p^2+1 primes are 10863652559262758117, 10863652743839069477, 10863652822943203397, 10863652954783427237, 10863653218463877317, 10863653271199967717

No solutions for k=7 until p < 35,981,630,539.

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