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}
}
  • C. Pomerance
  • Published 1 September 1981
  • Computer Science
  • The Mathematical Intelligencer
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… 

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References

SHOWING 1-10 OF 35 REFERENCES

On the exact number of primes less than a given limit

The problem of counting the exact number of primes x, wthout actually listing them all, dates from Legendre [l] who observed that the number of primes p for which x p x is one less than where [z]

Some algorithms for prime testing using generalized Lehmer functions

Let N be an odd integer thought to be prime. The properties of special functions which are generalizations of the functions of Lehmer (Ann. of Math., v. 31, 1930, pp. 419-448) are used to develop

Some observations on primality testing

Let N be an integer which is to be tested for primality. Previous methods of ascertaining the primality of N make use of factors of N ? 1, N2 ? N + 1, and N2 + 1 in order to increase the size of any

Riemann's Hypothesis and tests for primality

  • G. Miller
  • Computer Science, Mathematics
    STOC
  • 1975
TLDR
It is shown that primality is testable in time a polynomial in the length of the binary representation of a number, and a partial solution is given to the relationship between the complexity of computing the prime factorization of a numbers, computing the Euler phi function, and computing other related functions.

Some probabilistic remarks on Fermat's last theorem

Let a i < a2 < ' * * be an infinite sequence of integers satisfying ön = (c Ho(l))n a for some a > 1. One can ask: Is it likely that d{ + dj = ar or, more generally, aix + • • • + ain = aif has

Every Prime has a Succinct Certificate

  • V. Pratt
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1975
TLDR
It remains an open problem whether a prime n can be recognized in only $\log _2^\alpha n$ operations of a Turing machine for any fixed $\alpha $.

Miller's Primality Test

A new lower bound for the pseudoprime counting function

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

The least quadratic non residue

This, the problem of the least quadratic non residue, has often been investigated. The best result is due to Vinogradov, who proved that (1) n(k) = O(k1I(2Ve) where n(k) denotes the least positive