Primes in arithmetic progression: Difference between revisions

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'''Consecutive primes in arithmetic progression''' refers to at least three ''consecutive'' primes which are consecutive terms in an arithmetic progression. Note that unlike an AP-''k'', all the other numbers between the terms of the progression must be composite. For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime.

For an integer ''k'' ≥ 3, a '''CPAP-''k''''' is ''k'' consecutive primes in arithmetic progression. It is conjectured there are arbitrarily long CPAP's. This would imply infinitely many CPAP-''k'' for all ''k''. The middle prime in a CPAP-3 is called a [[balanced prime]]. The largest known {{as of|20182022|lc=on}} has 1054615004 digits.

The first known CPAP-10 was found in 1998 by Manfred Toplic in the [[distributed computing]] project CP10 which was organized by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.<ref>H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, P. Zimmermann, [https://www.ams.org/mcom/2002-71-239/S0025-5718-01-01374-6/home.html ''Ten consecutive primes in arithmetic progression''], [[Mathematics of Computation]] 71 (2002), 1323–1328.</ref> This CPAP-10 has the smallest possible common difference, 7# = 210. The only other known CPAP-10 as of 2018 was found by the same people in 2008.