Fermat primality test

Abstract: Upper bounds for the power of p which divides the Fermat quotient qa= (a 1 1)/p are obtained, and conditions are given which imply that q 0 (mod p). The results are in terms of the number of steps in a simple algorithm which determines the semiorder of a (mod p).

Abstract: A new prime factor is given for each of the Fermat numbers F12 and F13 (none was previously known for F13). The factoring method used and its machine implementation are discussed. A short table of factors and a current status list are also included. In recent years various investigators have used computers to search for prime factors of the Fermat numbers Fm = 22m + 1, m > 7 (see Selfridge [10], Robinson [7] -[9] , Riesel [6] , Brillhart [1], Wrathall [12] , Morrison and Brillhart [3], [4] ). In our investigation we have found two new factors, namely: 190274191361 = 11613415 * 214 + 1 and 2710954639361 = 41365885 . 216 + 1, which divide F1 2 and F13, respectively. Previously, three prime factors of F1 2 hac been discovered, while F1 3 was only known to be composite (see Paxson [5] ). It is well known that any prime factor of Fm has the form k 2m + 2 + 1, m > 2. In searching for such a factor, we can try dividing Fm by each dk of this form for k less than some search limit Lm. Many composite dk can, of course, be eliminated as trial divisors in advance by sieving on the arithmetic sequence {dk} with small, odd primes (in our program the odd primes less than 500 were used). To discover whether a dk which has survived the sieving is a factor of Fm, we calculate Fm(mod dk) by the usual powering method-here only a sequence of squarings and reductions. Since the residues ri of the powers 22'(mod dk), i m, then this factor can only be discovered during the search for factors of Fn. For example, the factor of F1 3 was found during the investigation of F14, since in this case n m = 1. This procedure was coded in COMPASS assembly language for the CDC 6400 at Received May 20, 1974. AMS (MOS) subject classifications (1970). Primary 10A25, 10A40; Secondary 10-04.