Factors Of Fermat Numbers Research Articles

Three new factors of Fermat numbers

We report the discovery of a new factor for each of the Fermat numbers F 13 , F 15 , F 16 These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively.

New factors of Fermat numbers

Forty-six new factors of Fermat numbers are given, along with a summary of the search limits.

New Factors of Fermat Numbers

Forty-six new factors of Fermat numbers are given, along with a summary of the search limits

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

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

Factors of Fermat numbers and large primes of the form 𝑘⋅2ⁿ+1

A new factor is given for each of the Fermat numbers F 52 , F 931 , F 6835 {F_{52}},{F_{931}},{F_{6835}} , and F 9448 {F_{9448}} . In addition, a factor of F 75 {F_{75}} discovered by Gary Gostin is presented. The current status for all F m {F_m} is shown in a table. Primes of the form k ⋅ 2 n + 1 , k k \cdot {2^n} + 1,k odd, are listed for 31 ⩽ k ⩽ 149 31 \leqslant k \leqslant 149 , 1500 > n ⩽ 4000 1500 > n \leqslant 4000 , and for 151 ⩽ k ⩽ 199 151 \leqslant k \leqslant 199 , 1000 > n ⩽ 4000 1000 > n \leqslant 4000 . Some primes for even larger values of n are included, the largest one being 5 ⋅ 2 13165 + 1 5 \cdot {2^{13165}} + 1 . Also, a survey of several related questions is given. In particular, values of k such that k ⋅ 2 n + 1 k\cdot {2^n} + 1 is composite for every n are considered, as well as odd values of h such that 3 h ⋅ 2 n ± 1 3h\cdot {2^n} \pm 1 never yields a twin prime pair.

Six New Factors of Fermat Numbers

Six New Factors of Fermat Numbers

Six new factors of Fermat numbers

A new prime factor is given for each of the Fermat numbers F 29 {F_{29}} , F 36 {F_{36}} , F 99 {F_{99}} , F 147 {F_{147}} , F 150 {F_{150}} , and F 201 {F_{201}} . A summary of search limits and recent results is included.

Some very large primes of the form 𝑘⋅2^{𝑚}+1

Several large primes of the form k ⋅ 2 m + 1 k \cdot {2^m} + 1 with 3 ⩽ k ⩽ 29 3 \leqslant k \leqslant 29 and m > 1500 m > 1500 are tabulated and four new factors of Fermat numbers are presented.

New primes of the form 𝑘⋅2ⁿ+1

All primes of the form k ⋅ 2 n + 1 k \cdot {2^n} + 1 for k odd, 1 ⩽ k > 150 1 \leqslant k > 150 , 1 ⩽ n ⩽ 1500 1 \leqslant n \leqslant 1500 , have now been computed. Those not previously published are given here. Numbers with 151 ⩽ k ⩽ 999 151 \leqslant k \leqslant 999 and n ⩽ 600 n \leqslant 600 were also tested. Three new factors of Fermat numbers and a large pair of twin primes were found.

Four new factors of Fermat numbers

Four new factors of Fermat numbers

Corrigenda to: “Two new factors of Fermat numbers” (Math. Comp. 29 (1975), 110)

Corrigenda to: “Two new factors of Fermat numbers” (Math. Comp. 29 (1975), 110)

Two New Factors of Fermat Numbers

Two New Factors of Fermat Numbers

Two new factors of Fermat numbers

A new prime factor is given for each of the Fermat numbers F 12 {F_{12}} and F 13 {F_{13}} (none was previously known for F 13 {F_{13}} ). The factoring method used and its machine implementation are discussed. A short table of factors and a current status list are also included.

New factors of Fermat numbers

From Table 1, in turn, we may compute [3], [4] the Hardy-Littlewood constants ha for a = =t5 and ? 10. Together with previously computed values, we may thus complete an 8D table of ha for a = -10(1)10 except for a = 47. The L+7(s), needed to fill this gap, may also be expressed in terms of I, (a) and R8( a), but this time the arguments a are not given explicitly in [2], and elaborate interpolation would be required to obtain comparable precision. Alternatively, as is known, generalized harmonic series, including La (s) for integer s, may be expressed in terms of the polygamma functions [5], [6]. However, the same difficulty arises for L?7(s), and again elaborate and laborious interpolation is necessary. At the author's request John W. Wrench, Jr. has kindly computed L7(2), L7(4), L-7(3) and L-7(5) in this way, and these numbers, together with the closed-form L?7(s), suffice to complete our tabulation of ha. This is given in Table 2.

A note on octic permutation polynomials

1. JOHN BRILLHART & G. D. JOHNSON, On the factors of certain Mersenne Y Math. Comp., v. 14, 1960, p. 365-369. 2. JOHN BRILLHART, Concerning the numbers 22P + 1, p prime, Math. Comp., v. 16, 1962, p. 424-430. (Reference is there made to the earlier table by M. Kraitchik.) 3. A. J. C. CUNNINGHAM & H. J. WOODALL, Factorizations of (yn =F 1), Hodgson, London, 1925, p. 19. 4. D. H. LEHMER, Tests for primality by the converse of Fermat's theorem, Bull., Amer. Math. Soc., v. 33, 1927, p. 327-340. 5. R. M. ROBINSON, A report on primes of the form k-2 + 1 and on factors of Fermat numbers, Proc., Amer. Math. Soc., v. 9, 1958, p. 673-681.

A Report on Primes of the Form k⋅2 n + 1 and On Factors of Fermat Numbers

A Report on Primes of the Form k⋅2 n + 1 and On Factors of Fermat Numbers

A report on primes of the form 𝑘⋅2ⁿ+1 and on factors of Fermat numbers

This note is a report on some calculations carried out during 1956 and 1957 on the SWAC, a high-speed computer located on the Los Angeles campus of the University of California, and supported in part by the Office of Naval Research. A preliminary report appeared in [11], and a few copies of a large table [12] were printed. The routines used on the SWAC were prepared by the author. My thanks are due to John L. Selfridge for assistance in operating the computer, and for valuable suggestions.

Note on the factors of Fermat’s numbers

Note on the factors of Fermat’s numbers