New Cullen primes

Keller, Wilfrid

doi:10.1090/S0025-5718-1995-1308456-3

New Cullen primes
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Math. Comp. 64 (1995), 1733-1741 Request permission

Abstract:

Numbers of the forms ${C_n} = n \cdot {2^n} + 1$ and ${W_n} = n\cdot {2^n} - 1$ are both called Cullen numbers. New primes ${C_n}$ are presented for $n = 4713,5795,6611,18496$. For ${W_n}$, several new primes are listed, the largest one having $n = 18885$. Furthermore, all efforts made to factorize numbers ${C_n}$ and ${W_n}$ are described, and the result, the complete factorization for all $n \leq 300$, is given in a Supplement.

References

Cullen numbers

8

Richard P. Brent, An improved Monte Carlo factorization algorithm, BIT 20 (1980), no. 2, 176–184. MR 583032, DOI 10.1007/BF01933190

Some integer factorization algorithms using elliptic curves

8

MVFAC: A vectorized Fortran implementation of the elliptic curve method

John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr., Factorizations of $b^n \pm 1$, 2nd ed., Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, RI, 1988. $b=2,3,5,6,7,10,11,12$ up to high powers. MR 996414, DOI 10.1090/conm/022

Question 15897

Solution of Question 15897

10

Factorisation of

and

47

Generalized Cullen numbers

21

G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. MR 1555183, DOI 10.1007/BF02403921
C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics, No. 70, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0404173
Ingemar Jönsson, On certain primes of Mersenne-type, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 117–118. MR 311558, DOI 10.1007/bf01932681

Prime factors of Cullen numbers

D. H. Lehmer, On the factors of $2^n\pm 1$, Bull. Amer. Math. Soc. 53 (1947), 164–167. MR 19635, DOI 10.1090/S0002-9904-1947-08774-7

Recent discoveries of large primes

6

Sur la série récurrente de Fermat

11

Peter L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp. 48 (1987), no. 177, 243–264. MR 866113, DOI 10.1090/S0025-5718-1987-0866113-7
Michael A. Morrison and John Brillhart, A method of factoring and the factorization of $F_{7}$, Math. Comp. 29 (1975), 183–205. MR 371800, DOI 10.1090/S0025-5718-1975-0371800-5

UBASIC: a Public-Domain BASIC for Mathematics

36

UBASIC Update

38

J. M. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc. 76 (1974), 521–528. MR 354514, DOI 10.1017/s0305004100049252
Hans Riesel, Lucasian criteria for the primality of $N=h\cdot 2^{n} -1$, Math. Comp. 23 (1969), 869–875. MR 262163, DOI 10.1090/S0025-5718-1969-0262163-1
Raphael M. Robinson, A report on primes of the form $k\cdot 2^{n}+1$ and on factors of Fermat numbers, Proc. Amer. Math. Soc. 9 (1958), 673–681. MR 96614, DOI 10.1090/S0002-9939-1958-0096614-7
Robert D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), no. 177, 329–339. MR 866119, DOI 10.1090/S0025-5718-1987-0866119-8
Ray P. Steiner, On Cullen numbers, BIT 19 (1979), no. 2, 276–277. MR 537790, DOI 10.1007/BF01930860
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms $M=(6a+1)2^{2m-1}-1$ and $M^{\prime } =(6a-1)2^{2m}-1$, Math. Comp. 22 (1968), 420–422. MR 227095, DOI 10.1090/S0025-5718-1968-0227095-2