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# Consecutive primes in arithmetic progression

It is conjectured that there are arithmetic progressions of ${\displaystyle \scriptstyle n\,}$ consecutive primes for any ${\displaystyle \scriptstyle n\,}$, but this has not even been (logically) proved for ${\displaystyle \scriptstyle n\,=\,3\,}$, although there are proofs by examples for ${\displaystyle \scriptstyle n\,}$ up to 10. It seems likely that there are infinitely many CPAP-${\displaystyle \scriptstyle n\,}$ with prime difference ${\displaystyle \scriptstyle c\cdot n\#\,}$, for all ${\displaystyle \scriptstyle c\,}$ and ${\displaystyle \scriptstyle n\,}$, where ${\displaystyle \scriptstyle n\#\,}$ is the primorial of ${\displaystyle \scriptstyle n\,}$ and CPAP stands for Consecutive Primes in Arithmetic Progression.[1]

## Occurrences of n CPAP

• In 1967, Jones, Lal and Blundon found five consecutive primes in arithmetic progression, A293791: (${\displaystyle \scriptstyle 10^{10}+24493+30k,\,k\,=\,0,...,4\,}$).[2]
• In 1967, Lander and Parkin found six consecutive primes in arithmetic progression: (${\displaystyle \scriptstyle 121174811+30k,\,k\,=\,0,...,5\,}$).[3]
• In 1995, Dubner and Nelson found seven consecutive primes in arithmetic progression:[4]
• In 1997, Dubner, Forbes, Lygeros, Mizony, Nelson and Zimmermann found eight consecutive primes in arithmetic progression:[4]
• In 1998, Toplic found nine consecutive primes in arithmetic progression:[5]
• In 1998, Toplic found ten consecutive primes in arithmetic progression, A033290.[5]

### First and smallest occurrence of n CPAP

The first and smallest occurrence of ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ consecutive primes in arithmetic progression are listed in A006560:

• a(1) = 2: (2) (degenerate arithmetic progression);
• a(2) = 2: (2, 3);
• a(3) = 3: (3, 5, 7);
• a(4) = 251: (251, 257, 263, 269); (not 5+k*6, because it skips 7,13,19)
• a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);[3]
• a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);[3]
• a(7) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k), k ≥ 210;[6]
• a(8) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k), k ≥ 210; The expected size is a(8) > ?.
• a(9) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k, ? + 8k), k ≥ 210; The expected size is a(9) > ?.
• a(10) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k, ? + 8k, ? + 9k), k ≥ 210; The expected size is a(10) > ?.

### Common differences of first and smallest occurrence of n CPAP

The common differences of first and smallest arithmetic progression of ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ consecutive primes are (Cf. A126989)

{0, 1, 2, 6, 30, 30, ≥ 210, ≥ 210, ≥ 210, ≥ 210, ≥ 2310, ≥ 2310, ≥ 30030, ...}
or {0, 1#, 2#, 3#, 4#, 4#, ≥ 5#, ≥ 5#, ≥ 5#, ≥ 5#, ≥ 6#, ≥ 6#, ≥ 7#, ...}

where a(7)=210 is unconfirmed and ${\displaystyle \scriptstyle n\#\,}$ is the ${\displaystyle \scriptstyle n\,}$th primorial listed in A002110.[7]