# Recent developments in primality testing

@article{Pomerance1981RecentDI, title={Recent developments in primality testing}, author={Carl Pomerance}, journal={The Mathematical Intelligencer}, year={1981}, volume={3}, pages={97-105} }

ConclusionIs there an algorithm that can decide whethern is prime or composite and that runs in polynomial time? The Adleman-Rumely algorithm and the Lenstra variations come so close, that it would seem that almost any improvement would give the final breakthrough.Is factoring inherently more difficult than distinguishing between primes and composites? Most people feel that this is so, but perhaps this problem too will soon yield.In his “Disquisitiones Arithmeticae” Gauss [9], p. 396, wrote“The…

## 65 Citations

### The Recognition of Primes

- Mathematics
- 2011

One very important concern in number theory is to establish whether a given number N is prime or composite. At first sight it might seem that in order to decide the question an attempt must be made…

### How to Recognize Whether a Natural Number is a Prime

- Mathematics
- 1996

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.

### Number theory and the real world

- Mathematics
- 1985

Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. According to Carl Friedrich Gauss, the "Princeps Mathemat icorum",…

### Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

- Computer ScienceMFCS
- 2022

This paper shows that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes, and proves that the Lenstra–Pomerance–Wagstaﬀ Conjecture implies that all O-ambiguity NP sets are in the restricted counting class RC PRIMES.

### The Reliability of Randomized Algorithms

- PhilosophyThe British Journal for the Philosophy of Science
- 2000

The prospects for establishing that randomized algorithms are reliable are analyzed and it is argued that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms.

### Implementazione di tests probabilistici di primalità

- PhysicsANNALI DELL UNIVERSITA DI FERRARA
- 1982

SuntoIn questo lavoro si considerano due tests probabilistici di primalità per interi disparim di forma qualsiasi. Gli algoritmi sono tali che sem è dichiarato composto allora lo è certamente, mentre…

### Open problems in number theoretic complexity, II

- MathematicsANTS
- 1994

This chapter presents a collection of 36 open problems in number theoretic complexity, showing how questions about the integers have natural generalizations to rings of integers in an algebraic number field, and questions about elliptic curves may generalize to arbitrary abelian varieties.

### Zassenhaus Conjecture on torsion units holds for PSL(2,p) with p a Fermat or Mersenne prime

- MathematicsJournal of Algebra
- 2019

### On the Stability of m-Sequences

- Mathematics, Computer ScienceIMACC
- 2011

We study the stability of m-sequences in the sense of determining the number of errors needed for decreasing the period of the sequences, as well as giving lower bounds on the k -error linear…

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In particular, we may take 5/14. ErdiSs conjectures that (x) x -)where e(x) 0 as x oo. See Pomeranee, Selfridge, Wagstaff [10] for more on this. Our main result holds for pseudoprimes to any base and…