On Conditional Probabilities of Factoring Quadratics

• 作者: Gabriela von Linden Thomas Beatty
• 来源期刊: 理论数学进展(英文)
• 学科: 数学
• 关键词: FACTORIZATION POLYNOMIAL QUADRATIC INTEGERS Rational Numbers Monic Modular ARITHMETIC CONDITIONAL Probability
• 年，卷（期）: 2020,（3）
• 页码范围: 114-124
• 页数: 11页
• 分类号: O17

摘要

Factoring quadratics over Z is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common. To the contrary, if coefficients of a general quadratic are selected randomly without restriction, the probability that a factorization exists is zero. We achieve a specific quantification of the probability of factoring quadratics by taking a new approach that considers the absolute size of coefficients to be a parameter n. This restriction allows us to make relative likelihood estimates based on finite sample spaces. Our probability estimates are then conditioned on the size parameter n and the behavior of the conditional estimates may be studied as the parameter is varied. Specifically, we enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter. The conditional probability of factorization as a function of n is just the ratio of this enumeration to the total number of possible quadratics consistent with n. This approach is patterned after the well-known case where factorizations are carried out over a finite field. We review the finite field method as background for our method of dealing with Z [x]. The monic case is developed independently of the general case because it is simpler and the resulting probability estimating formula is more accurate. We conclude with a comparison of our theoretical probability estimates with exact data generated by a computer search for factorable quadratics corresponding to various parameter values.