Factors of Fermat Numbers and Large Primes of the Form k ⋅2 n + 1

The factorization of the ninth Fermat number

“…The many results on factors of Fermât numbers that have been obtained by the methods above, as well as bibliographic information, can be found in [17,Chapter XV;16,7,41,23]. For up-to-date information one should consult the current issues of Mathematics of Computation, as well as the updates to [7] that are regularly published by S. S. Wagstaff, Jr. We give a brief summary of the present state of knowledge.…”

The Twenty-Second Fermat Number is Composite

“…Aspiring factorers should know that factors for the midrange, say, F\q through F14, have been fairly well weeded out by applications of ECM, in the sense that there are probably no more hidden factors in this range possessed of less than thirty digits (but one cannot be completely sure yet-the observation is merely statistically motivated). A factorer should also note the sieving limits, as reported in [3], indicating that, in the higher range « = 18 -22, hidden factors (k2"+2 + 1) have been ruled out for k < 236. One might therefore summarize the current factoring status as follows: Direct sieving is a nearly exhausted option, the ECM may have just a little potential left (e.g., for the upper regions of Table 2), while the NFS seems hard to apply at any higher levels n > 9.…”

The General Sinus

“…Even today no other Fermat primes have been found, although the smallest unsettled cases are F 22 , F 24 and F 26 (The primarily status of Fermat numbers as of 1983 can be found in Keller [15] and a shorter but more recent table is included in Young and Buell [16]). …”