Siemion Fajtlowicz has been promoting the topic of $p$-long arithmetic progressions of primes which start with $p$ during 1991-4 (or longer). A friend of his got an $11$-long progression. Then, soon after, I got a small theorem which allowed me to get bunches of such $11$-progressions very fast, and also a lot of $13$-progressions (of $13$ primes starting with $13$) almost as quickly.

On the other hand, I conjectured that only a finite number of primes $p$ start $p$-progressions. Moreover, I believe that no prime $p>13$ starts any. Perhaps there is already none for $\ p=17$. (I got my results during 1991 or 1992).

Siemion Fajtlowicz has been promoting the topic of $p$-long arithmetic progressions of primes which start with $p$ during 1993-4 (or longer). He and his colleague Micha Hofri got an $11$-long progression. Then, soon after, I got a small theorem which allowed me to get bunches of such $11$-progressions very fast, and also a lot of $13$-progressions (of $13$ primes starting with $13$) almost as quickly.

On the other hand, I conjectured that only a finite number of primes $p$ start $p$-progressions. Moreover, I believe that no prime $p>13$ starts any. Perhaps there is already none for $\ p=17$. (I got my results during 1994).

See also: http://primerecords.dk/aprecords.htm