This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) . 1.

. We introduce an algorithm that computes the prime numbers up to N using O(N=log log N) additions and N 1=2+o(1) bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms. 1.

Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an x-pseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.

. An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: -- it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for fixed n; -- the transition from n to n + 1 takes only O(1) additions of numbers of size O(n). (On average, of course, O(1) such additions increase the limit up to which all primes are known from n to n + \Theta(log log n)). 1 Introduction A so-called "formula" for the i'th prime has been a long-lived concern, if not quite the Holy Grail, of Elementary Number Theory. This concern seems poorly motivated, as evidenced by the extraordinary freak-show of solutions proffered over the ages. The natural setting is Algorithmic Number Theory, and what is desired is much better cast as an algorithm to compute the i'th prime. Given that app...

. Let d 0 (n) = pn , the n-th prime, for n 1, and let d k+1 (n) = jd k (n) \Gamma d k (n + 1)j for k 0, n 1. A well known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19-th century, says that d k (1) = 1 for all k 1. This paper reports on a computation that verified this conjecture for k ß(10 13 ) ß 3 \Theta 10 11 . It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences. 1. Introduction Let p 1 = 2, p 2 = 3; : : : be the primes in their natural ordering, and set d 0 (n) = p n ; n 1 d k+1 (n) = jd k (n) \Gamma d k (n + 1)j ; k 0; n 1 : (1.1) Table 1 shows d k (n) for 0 k 20, 1 n 20. Note that d k (1) = 1 for 1 k 20. As was pointed out by H. C. Williams, Proth [15] claimed to prove that d k (1) = 1 for all k 1, but his proof was faulty. More recently, Gilbreath (unpublished) independently conjectured that d k (1) = 1 for all k ...

. A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O( p n=(log log n) 2 ) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l ? 1 is constant, and two super-linear time sieves that use very little space. 1 Introduction A prime number sieve is an algorithm that finds all prime numbers up to a bound n. In this paper we present four new prime number sieves, three of which accept a parameter to control their use of time versus space. The fastest known prime number sieve is the dynamic wheel sieve of Pritchard [11], which uses O(n= log log n) arithmetic operations and O(n= log log n) bits of space. Dunten, Jones, and Sorenson [6] gave an algorithm with the same asymptotic running time, while using only O(n=(log ...

Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, space-efficient implementation of the wheel datastructure. 1