Are there primes in every arithmetic progression? If so, how many?  Dirichlet’s theorem tells that the answers are usually ‘yes,’ and ‘there are infinitely many primes. ‘Dirichlet’s Theorem on Primes in Arithmetic Progressions (1837)If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b, … contains infinitely many primes. This theorem does not say that there are infinitely may consecutive terms in this sequence which are primes. First van der Corput (in 1939) and then Chowla (in 1944) proved this for the case of three consecutive terms.  Finally, in 2004, Ben Green and Terence Tao proved that there were arbitrarily long arithmetic progressions of primes. Here though we have an even more stringent condition. We are looking for n consecutive primes in arithmetic progressions.  It is conjectured that there are such primes, but this has not even been shown in for the case of n=3 primes. In 1967, Jones, Lal & Blundon found five consecutive primes in arithmetic progression: (1010 + 24493 + 30k, k = 0, 1, 2, 3, 4).  That same year Lander & Parkin discovered six (121174811 + 30k, k = 0, 1, …, 5).  After a gap of twenty years the number was increased from six to seven by Dubner & Nelson; then in quick succession, eight, nine and finally ten by Dubner, Forbes, Lygeros, Mizony, Nelson & Zimmermann.

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