Large Prime Numbers

   For about seventy-five years the largest known
prime number has been P = 2^127 - 1, identified as
such by Edouard Lucas.  It has remained the largest
known in spite of many attempts to identify larger
ones, although there have been conjectures and claims
insufficiently substantiated or entirely unproved.
   Recently we have prepared a routine for testing
on the Edsac the primality of numbers of the form
kP + 1, with P as defined above, and have found
ten values of k that give prime numbers.  The largest
of these gives the present (June 7) largest known
prime, namely,
                934(2^127 - 1) + 1
                                   J. C. P. Miller
June 7.                            D. J. Wheeler
   Note added in proof (October 8). Further work on
the Edsac by Wheeler and myself has demonstrated
the primality of
                978(2^127 - 1) + 1
and culminated in early July in the identification of
the present largest known prime,
              180(2^127 - 1 )^2 + 1.
   Also, in early July, A. Ferrier, of France, using a
desk machine, demonstrated the primality of
                  (2^148 + 1)/17,
which is the second largest known prime.
                                   J. C. P. MILLER
University Mathematical Laboratory,
            Cambridge.