Riemann's Hypothesis and tests for primality

@article{Miller1975RiemannsHA,
  title={Riemann's Hypothesis and tests for primality},
  author={Gary L. Miller},
  journal={Proceedings of the seventh annual ACM symposium on Theory of computing},
  year={1975}
}
  • G. Miller
  • Published 5 May 1975
  • Computer Science, Mathematics
  • Proceedings of the seventh annual ACM symposium on Theory of computing
The purpose of this paper is to present new upper bounds on the complexity of algorithms for testing the primality of a number. The first upper bound is 0(n1/7); it improves the previously best known bound of 0(n1/4) due to Pollard [11]. The second upper bound is dependent on the Extended Riemann Hypothesis (ERH): assuming ERH, we produce an algorithm which tests primality and runs in time 0((log n)4) steps. Thus we show that primality is testable in time a polynomial in the length of the… Expand
Riemann's hypothesis and tests for primality
In this paper we present two algorithms for testing primality of an integer. The first algorithm runs in 0(n1/7) steps; while, the second runs in 0(log4n) step but assumes the Extended RiemannExpand
Explicit bounds for primality testing and related problems
Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omitsExpand
Combinatorial primality test
In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If anExpand
Miller's Primality Test
  • H. Lenstra
  • Mathematics, Computer Science
  • Inf. Process. Lett.
  • 1979
Theorem 1. Assume that for every integer d that is 1 mod 4 and either prime or the product of two primes, the L-function I= & (k/d) l kmS satisfies the generalized Riemann hypothesis, where {kid)Expand
Primality and identity testing via Chinese remaindering
We give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over Zn holds.We also give new randomizedExpand
Conditional bounds for the least quadratic non-residue and related problems
TLDR
The existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression are improved and the classical conditional bounds of Littlewood for L-functions at s=1 are refined. Expand
Pseudopowers and primality proving
TLDR
A generalization of the result for any odd prime r is presented, 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. Expand
THE LEAST QUADRATIC NON-RESIDUE, VALUES OF L-FUNCTIONS AT s = 1, AND RELATED PROBLEMS
In this paper, we study explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specically, we improve the existingExpand
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 anyExpand
Algebraic Geometry Over Four Rings and the Frontier to Tractability
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity ofExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 21 REFERENCES
The distribution of quadratic and higher residues
In this paper we discuss some of the many problems that can be propounded concerning the distribution of the quadratic residues and non-residues, or more generally the kth power residues andExpand
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean valueExpand
The complexity of theorem-proving procedures
  • S. Cook
  • Computer Science, Mathematics
  • STOC
  • 1971
It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is aExpand
The distribution of quadratic residues and non-residues
If p is a prime other than 2, half of the numbers 1, 2, … p —1 are quadratic residues (mod p ) and the other half are quadratic non-residues. Various questions have been proposed concerning theExpand
Reducibility among combinatorial problems" in complexity of computer computations
TLDR
In his 1972 paper, Reducibility Among Combinatorial Problems, Richard Karp used Stephen Cooks 1971 theorem that the boolean satisfiability problem is. Expand
The Art of Computer Programming
TLDR
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. Expand
Reducibility Among Combinatorial Problems," Complexity of Computer Computations, R.E. Miller and 3.W
  • 1972
Schnelle Multiplikation Grosser Zahlen
  • Computing
  • 1971
...
1
2
3
...