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- W1481471444 abstract "Several large primes of the form k ■ 2m + 1 with 3 1500 are tabulated and four new factors of Fermât numbers are presented. It is well known that any factor of the Fermât number Fn = 22 +1 must have the form k ■ 2m + 1 where m > n + 2 and k is odd. Baillie (1) has extended the earlier tables of Robinson (6) and Matthew and Williams (5) to include all primes of the above form for odd k < 149 and m < 1500. Only 25 of these primes are factors of Fermât numbers, and of these, 21 have k < 29. In this note, we describe the results of searching for all primes of the form k ■ 2m + 1 with k < 29 and m < r. Here r is at least 4,000 and, in certain special cases, is as large as 8,000 or 10,000. The numbers to be tested for primality were first sieved by solving the congruence 2m = -A:-1 (mod p) for all primes p less than 4 x IO6. This was done by making use of a modification of an algorithm mentioned by Knuth (4, p. 9). All values of 2m (mod p) were computed for 0 < m < 100 and stored in a table; next, all values of -k~i2~100 (mod p) (0 < n < 100) were computed and looked up in the table by hashing. When a match was found, k2m + 100n + 1 was known to be divisible by p. This preliminary sieving tech- nique eliminated about 90% of all the numbers. The remaining numbers were tested for primality by using the test of Proth; see Robinson (6). Since the numbers involved in this testing are very large, the algorithm of recursive bisection (Knuth (3, p. 258)) was used to increase the speed of multipli- cation. This algorithm allows two n-bit numbers to be multiplied in three %n bit multi- plications, provided n is a power of 2. For n = 8192, this technique is 4.8 times faster than the usual multiplication algorithm. Also, advantage was taken of the special form of the numbers in order to reduce the problem of division by k ■ 2m + 1 to that of division by k. The results of our computations are presented in Table 1 below. These calculations were performed in over 100 CPU hours on an AMDAHL 470-V7 computer. The upper bound on the range of m was increased beyond 4000 when it was felt that the density of Fermât number factors in the sequence {k2m + 1, m = 2, 3, . . . , 4000} was large enough that there was a good chance of finding another such factor. Four of these primes are divisors of Fermât numbers. They are:" @default.
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- W1481471444 date "1980-01-01" @default.
- W1481471444 modified "2023-09-26" @default.
- W1481471444 title "Some very large primes of the form $k·2sp{m}+1$" @default.
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- W1481471444 doi "https://doi.org/10.1090/s0025-5718-1980-0583519-8" @default.
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