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 = 4p^{2} + 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) 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". *** Giovanni Resta wrote (Nov. 2004): Contrary to what Russo wrote, there exists a
solution for 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. ***





