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A Survey of Modern Integer Factorization Algorithms
 CWI Quarterly
, 1994
"... Introduction An integer n ? 1 is said to be a prime number (or simply prime) if the only divisors of n are \Sigma1 and \Sigman. There are infinitely many prime numbers, the first four being 2, 3, 5, and 7. If n ? 1 and n is not prime, then n is said to be composite. The integer 1 is neither prime ..."
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Introduction An integer n ? 1 is said to be a prime number (or simply prime) if the only divisors of n are \Sigma1 and \Sigman. There are infinitely many prime numbers, the first four being 2, 3, 5, and 7. If n ? 1 and n is not prime, then n is said to be composite. The integer 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a finite (perhaps empty) product of prime numbers, and that this factorization is unique except for the ordering of the factors. Table 1.1 has some sample factorizations. 1990 = 2 \Delta 5 \Delta 199 1995 = 3 \Delta 5 \Delta 7 \Delta 19 2000 = 2 4 \Delta 5 3 2005 = 5 \Delta 401
POLYNOMIALS OYER Q
, 1999
"... The Lucas sequences, which Include the Fibonacci numbers as a special case, arise as solutions to the recursion relation yn+i=®yn+byn_h «>1, (1) where a, b and (yn)n>o take values in some specified ring and a and b are fixed elements which do ..."
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The Lucas sequences, which Include the Fibonacci numbers as a special case, arise as solutions to the recursion relation yn+i=®yn+byn_h «>1, (1) where a, b and (yn)n>o take values in some specified ring and a and b are fixed elements which do
NOTE ON FIBONACCI PRIMALITY TESTING
, 1996
"... One of the most effective ways of proving an integer N is prime is to show first that N is a probable prime, i.e., that a N ~ l = 1 (mod N) for some base a and l<a<Nl, and then to find enough prime factors of N±l so that certain other conditions are satisfied (see [1] for details of such prim ..."
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One of the most effective ways of proving an integer N is prime is to show first that N is a probable prime, i.e., that a N ~ l = 1 (mod N) for some base a and l<a<Nl, and then to find enough prime factors of N±l so that certain other conditions are satisfied (see [1] for details of such primality tests). The problem of finding these prime factors is, of course, the difficult and
CONJECTURES ON THE ZDENSITIES OF THE FIBONACCI SEQUENCE
, 1996
"... The concept of "Zdensities " is introduced in this paper, leading to several interesting conjectures involving the divisibility properties of the Fibonacci entrypoint function. We let 2£ = {^}^=i and X = {Ln}™=l denote the Fibonacci and Lucas sequences, respectively. Given m, the Fibonac ..."
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The concept of "Zdensities " is introduced in this paper, leading to several interesting conjectures involving the divisibility properties of the Fibonacci entrypoint function. We let 2£ = {^}^=i and X = {Ln}™=l denote the Fibonacci and Lucas sequences, respectively. Given m, the Fibonacci entrypoint ofm, denoted by Z(m), is the smallest n> 0 such that m\Fn\
ON A HOGGATTBERGUM PAPER WITH TOTIENT FUNCTION APPROACH FOR DIVISIBILITY AND CONGRUENCE RELATIONS
, 1988
"... During their discussion of divisibility and congruence relations of the Fibonacci and Lucas numbers, Hoggatt & Bergum found values of n satisfying the congruences Fn E 0 (mod ft) or Ln E 0 (mod ft). In this connection, Hoggatt & Bergum's research appears in Theorems 1, 3, 5, 6, and 7 of ..."
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During their discussion of divisibility and congruence relations of the Fibonacci and Lucas numbers, Hoggatt & Bergum found values of n satisfying the congruences Fn E 0 (mod ft) or Ln E 0 (mod ft). In this connection, Hoggatt & Bergum's research appears in Theorems 1, 3, 5, 6, and 7 of [4]. The present paper originated on the same lines in search of values of n that satisfy <$>(Fn) E 0 (mod ft) or (j)(Ln) E 0 (mod ft), where § is the totient function. Before going into the analysis of the problem, we state some results that will be quoted frequently.
On the primality of F 4723 and F 5387
, 1999
"... Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], ..."
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Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], one has F 4k+3 \Gamma 1 = F k+1 L k+1 L 2k+1 : (1) Here k = 1180, k + 1 = 1181 and 2k + 1 = 2361 = 3 \Theta 787. From [3] and with the help of factors found by Montgomery and Silverman [7, 8, 9], we get F 1181 = 5453857 \Theta C 240 ; L 1181 = 59051 \Theta<F27.43