Prime sieves using binary quadratic forms

Prime sieves using binary quadratic forms

We introduce an algorithm that computes the prime numbers up to N using O(N/log logN) additions and N1/2+o(1) bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms.

Representation of Prime Powers by Binary Quadratic Forms

In this article, we consider the representation of prime powers by binary quadratic forms of discriminant D = −2q1 . . . qt where the product of primes q1 . . . qt ≡ 3 (mod 4), for instance if it is of special RichaudDegert type n2 ± 2 for odd n’s, n2 − 1 for even n’s. We consider all the ambiguous classes and Q( √|D0|), where D0 is the fundamental discriminant and we obtain a general criterion...

Division and Binary Quadratic Forms

has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...

On Perfect Binary Quadratic Forms

A quadratic form f is said to be perfect if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all perfect binary integer quadratic forms. If there is an integer bilinear map s such that f(s(x, y)) = f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f is perfect. We give an explicit descri...

Composition of Binary Quadratic Forms