On Pseudopoints of Algebraic Curves

Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod 8$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le x$, yet is not a square. We extend this defintion to algebraic curves and say that $n$ is an $x$-pseudopoint of a curve $f(u,v) = 0$ (where $f \in \Z[U,V]$) if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0 \pmod p$ is satisfied for some $m$. We use the Bombieri bound of exponential sums along a curve to estimate the smallest $x$-pseudopoint, which shows the limitations of the modular approach to searching for points on curves

Sieving for pseudosquares and pseudocubes in parallel using doubly-focused enumeration and wheel datastructures

We extend the known tables of pseudosquares and pseudocubes, discuss the implications of these new data on the conjectured distribution of pseudosquares and pseudocubes, and present the details of the algorithm used to do this work. Our algorithm is based on the space-saving wheel data structure combined with doubly-focused enumeration, run in parallel on a cluster supercomputer

The Pseudosquares Prime Sieve

We present the pseudosquares prime sieve, which finds all primes up to n

On Pseudosquares and Pseudopowers

Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\equiv1\pmod 8$ that is a quadratic residue for each odd prime $p\le x$, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most $\exp(a_g x/\log x)$ for a suitable constant $a_g$. A bound of $\exp(a_g x\log\log x/\log x)$ is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares

Pseudopowers and primality proving

It has been known since the 1930s that so-called pseudosquares yield a very powerful machinery for the primality testing of large integers N. In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2^80) this gives a deterministic primality test in time O((lg N)^(3+o(1))), which many believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be extended to pseudo r-th powers. Very recently, this was accomplished for r=3. In fact, the results obtained indicate that r=3 might lead to an even more powerful algorithm than r=2. This naturally leads to the challenge if and how anything can be achieved for r>3. The extension from r = 2 to r = 3 relied on properties of the arithmetic of the Eisenstein ring of integers Z[\zeta_3], including the Law of Cubic Reciprocity. In this paper we present a generalization of our result for any odd prime r. The generalization is obtained by studying the properties of Gaussian and Jacobi sums in cyclotomic ring of integers, which are tools from which the r-th power Eisenstein Reciprocity Law is derived, rather than from the law itself. While r=3 seems to lead to a more efficient algorithm than r=2, we show that extending to any r>3 does not appear to lead to any further improvements

The pseudosquares prime sieve

Abstract. We present the pseudosquares prime sieve

The I/O Complexity of Computing Prime Tables

International audienceWe revisit classical sieves for computing primes and analyze their performance in the external-memory model. Most prior sieves are analyzed in the RAM model, where the focus is on minimizing both the total number of operations and the size of the working set. The hope is that if the working set fits in RAM, then the sieve will have good I/O performance, though such an outcome is by no means guaranteed by a small working-set size. We analyze our algorithms directly in terms of I/Os and operations. In the external-memory model, permutation can be the most expensive aspect of sieving, in contrast to the RAM model, where permutations are trivial. We show how to implement classical sieves so that they have both good I/O performance and good RAM performance, even when the problem size N becomes huge—even superpolynomially larger than RAM. Towards this goal, we give two I/O-efficient priority queues that are optimized for the operations incurred by these sieves