Járai's prime hunting methods reloaded
(the largest known Cunningham chain of length \(2\)
of the \(2\)nd kind)

Abstract. Antal Járai established three research groups that dealt with computational number theory. Due to his brilliant ideas these teams were very successful. Moreover, he implemented the world's fastest arithmetic routines. He focused mainly on large prime combinations and his teams reached 19 world records from 1992 to 2014, namely they set the record for the largest known twin primes 9 times and Sophie Germain primes 7 times and a Cunningham chain of length 3 of the first kind. Furthermore they proved the primality of the largest known number of the form \(n^4 + 1\) and a number which is simultaneously twin and Sophie Germain prime. In this paper, we report on a new project proving that Járai's methods and routines are cutting edge tools for effective manipulation of large numbers even in 2020. We are celebrating Prof. Járai's 70th birthday with his 20th world record.