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Probabilistic Number Theory Project Description Prerequisites Warm up Problems Hints Resources Teaching Notes Extension Problems Results

Prerequisites for the
Probabilistic Number Theory Problem

This project, more than some of the others, may require you to learn some new background material. That often happens in research: starting a new problem requires some prep-time getting to know the territory.

You’ll need to be comfortable with the notion of prime number (we’ll often say “number” and mean “positive integer”). In later parts of the investigation, you’ll need to know the fundamental theorem of arithmetic: every number can be factored in one and only one way into a product of primes. While you don’t need to be able to prove this, several students who worked on the project took some time out to understand a proof, and they were glad they did. See The Fundamental Theorem of Arithmetic for more information.

The project can be approached on both an empirical level, gathering and analyzing data, and a theoretical level, making a mathematical model and proving conjectures.

An empirical investigation requires that you know the definition of probability: when the number of outcomes is finite, it is the number of favorable outcomes divided by the number of outcomes. It would also help if you are familiar with a software environment or a programming language that allows you to construct simulations and test integers for properties like being prime.

In addition, for the theoretical investigation, you’ll need to know about

  • limits (facts like  lim
	n-->o o 1
	n = 0)
  • geometric series and other infinite series
  • some calculus (integration by parts, integration as area)
  • a little trig (addition formulas for sine and cosine)

This sounds formidable, but if you’ve had a good trig course, your teacher or mentor can bring you up to speed on the rest.

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