Abstract. We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Abstract. Let ψm be the smallest strong pseudoprime to the first m prime bases. This value is known for 1 ≤ m ≤ 11. We extend this by finding ψ 12 and ψ 13 . We also present an algorithm to find all integers n ≤ B that are strong pseudoprimes to the first m prime bases; with a reasonable heuristic assumption we can show that it takes at most B 2/3+o(1) time.
We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW challenge pseudoprimes with two or three prime factors up to 2 80 . In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods.
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 .
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