Two Algorithms to Find Primes in Patterns

@article{Sorenson2020TwoAT,
  title={Two Algorithms to Find Primes in Patterns},
  author={J. Sorenson and Jonathan Webster},
  journal={Math. Comput.},
  year={2020},
  volume={89},
  pages={1953-1968}
}
Let $k\ge 1$ be an integer, and let $P= (f_1(x), \ldots, f_k(x) )$ be $k$ admissible linear polynomials over the integers, or \textit{the pattern}. We present two algorithms that find all integers $x$ where $\max{ \{f_i(x) \} } \le n$ and all the $f_i(x)$ are prime. Our first algorithm takes at most $O_P(n/(\log\log n)^k)$ arithmetic operations using $O(k\sqrt{n})$ space. Our second algorithm takes slightly more time, $O_P(n/(\log \log n)^{k-1})$ arithmetic operations, but uses only $n^{1/c… 
1 Citations
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References

SHOWING 1-10 OF 42 REFERENCES
Dense Admissible Sets
TLDR
The behavior of ρ *(x), in particular, the point at which ρ (x) first exceeds π (x), and its asymptotic growth is examined.
Strong pseudoprimes to twelve prime bases
TLDR
An algorithm to find all integers $n\le B$ that are strong pseudoprimes to the first $m$ prime bases is presented; with a reasonable heuristic assumption it is shown that it takes at most $B^{2/3+o(1)}$ time.
The Pseudosquares Prime Sieve
TLDR
The pseudosquares prime sieve is presented, which finds all primes up to n in sublinear time using very little space and the primes generated by the algorithm are proven prime unconditionally.
Dissecting a Sieve to Cut Its Need for Space
  • W. Galway
  • Computer Science, Mathematics
    ANTS
  • 2000
TLDR
A “dissected” sieving algorithm which enumerates primes in the interval [x 1, x 2], using \(O(x_{2}^{1/3})\) bits of memory and using arithmetic operations on numbers of \(O(\rm ln \it x_{2}\) bits.
Integers free of small prime factors in arithmetic progressions *
  • T. Xuan
  • Mathematics
    Nagoya Mathematical Journal
  • 2000
For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve,
An improved sieve of Eratosthenes
TLDR
The sieve of Eratosthenes will be able to use it to factor integers, and not just to produce lists of consecutive primes, and also has close ties to Voronoi's work on the Dirichlet divisor problem.
Statistical evidence for small generating sets
TLDR
This work gives additional evidence, independent of the ERH, that primality testing can be done in deterministic polynomial time; if the bound on G(n) is correct, there is a deterministic primality test using O(log n)2 multiplications modulo n.
A heuristic asymptotic formula concerning the distribution of prime numbers
Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these
Prime numbers and computer methods for factorization
1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning
Doubly focused enumeration of locally square polynomial values
Let f be a nonconstant squarefree polynomial. Which of the values f(c + 1), f(c + 2), . . . , f(c + H) are locally square at all small primes? This paper presents an algorithm that answers this
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4
5
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