PUMaC 2014 · 加试 · 第 6 题

PUMaC 2014 — Power Round — Problem 6

(2) Show that every positive integer can be represented by f . 2 2 In fact, it can be shown that more than 40 forms of the form ax + by + 2 2 cz + dw can represent all nonnegative integers - we investigate this further in subsequent sections. 5 Ternary Conic Polynomial Now that we have seen an example of a universal positive-definite conic polyno- mial, we dig deeper into analyzing conic polynomials. Much has been discussed about polynomials of two variables, and you may have noticed that two-variable conic polynomials represent a small subset of the positive integers. In the three- variable case, however, it is different – it represents a substantial portion of the positive integers. Let’s see how it goes. Definition 15. A ternary conic polynomial is a conic polynomial f ( x, y, z ) with three variables. Much of the facts about ternary conic polynomials is not elementary - it requires extensive use of theories beyond the number theory most of you know. Everything here, with a bit of hints tho, are approachable with elementary number theory. Good luck! For this section and this section ONLY, you may use the following theorem to your advantage: Theorem 16 (Dirichlet’s Theorem on Arithmetic Progressions) . If a, b are positive integers such that ( a, b ) = 1 , there are infinitely many primes of the form an + b where n ∈ Z . 8 2 2 2 We first look at the ”basic” ternary form x + y + z .

解析

暂无解答链接。

No solutions link available.