Some Primes of the Form (a n - 1)/(a - 1)

The primality of 𝑅1031

“…Now (IO103 -l)/9 is completely factored in [16] and the complete factorization of 10103 + 1 can be obtained from Brillhart et al [4] and results of Atkin and Rickert mentioned at the end of Williams [12]. Also (10515 + 1)/(10103 + 1) has 7211 • 9091…”

“…In [16] a search for further primes of this form for all n < 2000 turned up only one further candidate for primality, Ä1031. Unfortunately, the methods of primality testing available at the time were not adequate to prove Ä1031 a prime.…”

“…Since the publication of [16] a great deal of research has been devoted to the problem of primality testing. The very important results of Adelman, Pomerance, and Rumely [1] have been modified by Cohen and Lenstra [5] and have resulted in the tests described by Cohen and Lenstra [6].…”

“…• 497491 • 569836171 • 2013681931 as a factor and T = (10515 -1)/(10103 -1) has 41 • 271 • 5905014721 as a factor (see [16]). With this information we have a completely factored part F of N -1, where F is approximately 2.73 X 10251.…”

Generalized repunit primes

“…In [3], a truly prodigious amount of work has gone into factoring numbers of the form b" ±1 for b from 3 to 12 and values of « up to about 300. In [8], Williams and Seah tabulated all the generalized repunits that are prime or probable prime for b from 3 to 12 and n up to 1000…”