Doubly-Focused Enumeration of Pseudosquares and Pseudocubes

Wooding, Kjell; Williams, H. C.

doi:10.1007/11792086_16

Cited by 7 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2) Does it make sense to use Bernstein's doubly-focused enumeration to attempt to further reduce the running time? See [5,28,30] (3) A natural extension to our algorithms here is to allow the linear polynomials f i to potentially be higher degree, irreducible polynomials. See Schinzel's Hypothesis H (See [26] and [7, §1.2.2]) and the Bateman-Horn conjecture [4].…”

Section: Discussion and Future Workmentioning

confidence: 99%

Two algorithms to find primes in patterns

Sorenson

Webster

2020

Math. Comp.

View full text Add to dashboard Cite

Let k ≥ 1 be an integer, and let P = (f1(x), . . . , f k (x)) be k admissible linear polynomials over the integers, or the pattern. We present two algorithms that find all integers x where max {fi(x)} ≤ n and all the fi(x) are prime.• Our first algorithm takes at most OP (n/(log log n) k ) arithmetic operations using O(k √ n) space. • Our second algorithm takes slightly more time, OP (n/(log log n) k−1 ) arithmetic operations, but uses only exp O(log n/ log log n) space. This result is unconditional for k > 6; for 2 < k ≤ 6, the proof of its running time, but not the correctness of its output, relies on an unproven but reasonable conjecture due to Bach and Huelsbergen. We are unaware of any previous complexity results for this problem beyond the use of a prime sieve.We also implemented several parallel versions of our second algorithm to show it is viable in practice. In particular, we found some new Cunningham chains of length 15, and we found all quadruplet primes up to 10 17 .

show abstract

Section: Discussion and Future Workmentioning

confidence: 99%

Two algorithms to find primes in patterns

Sorenson

Webster

2020

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…We found the following new pseudocubes (only listed for p ≡ 1 (mod 3)): These pseudocubes were found in about 6 months of total wall time in 2009. Wooding and Williams [12] had found a lower bound of L 499,3 > 1.45152 × 10 22 . For a complete list of known pseudocubes, see [12,4,10].…”

Section: Introductionmentioning

confidence: 94%

“…This, then, motivates the search for larger and larger peudosquares and pseudocubes, and attempts to predict their distribution. See, for example, Wooding and Williams [12] and also [7,11,8,2,10].…”

Section: Introductionmentioning

confidence: 99%

Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures

Sorenson

2010

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…5 An imaginary quadratic field with class group exponent ≤ 8 not listed has smallest split prime > 193. 6 Searching for imaginary quadratic fields with |D| up to a given bound and smallest split prime > 193 can be done by a multiply-focused enumeration similar to [14]. I.e.…”

Section: Remarksmentioning

confidence: 99%