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Obviously, if there is an efficient algorithm for IFP, then by recursively executing the algorithm, the PFP can be efficiently solved. Thus, IFP and PFP can be regarded as computationally equivalent problems: P IFP ⇐⇒ PFP. , Bill Gates, who mentioned in this book The Road Ahead (page 265 in [117]) that The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers. In prime factorization, we do not factor a prime, but a composite, and in fact, we are trying to find prime factors from a large composite.

Computational/Mathematical Preliminaries [3] (Main Computation) Set 1 , xi+1 ← x − qi i i ← i + 1, qi ← xi , print(qi ), go to Step [2]. [4] Exit. 7. Let x = 160523347/60728973. 4, we get 160523347/60728973 = [2, 1, 1, 1, 4, 12, 102, 1, 1, 2, 3, 2, 2, 36]. 9. The continued fraction algorithm is in P. 5. If each qi is an integer, the continued fraction is called simple; a simple continued fraction can either be finite or infinite. A continued fraction formed from [q0 , q1 , q2 , · · · qn−1 , qn ] by neglecting all of the terms after a given term is called a convergent of the original continued fraction.

It is conjectured but not yet proven that: let π2 (x) be the number of primes p such that if p ≤ x is prime, and p + 2 is also prime, then (3-1) A weak form: There are infinitely many twin primes. That is, lim π2 (x) = ∞. 35) then lim x→∞ π2 (x) = 1. 36) Using very complicated arguments based on sieve methods, the Chinese mathematician J. R. Chen showed that there are infinitely many pairs of integers (p, p + 2), with p prime and p + 2 a product of at most two primes. (4) Mersenne Prime Conjecture: Mersenne primes are primes of the form 2p − 1, where p is prime.

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