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## Category:

## Description:

This course covers several topics from number theory, and assumes only basic knowledge (Euclid's algorithm, prime numbers, etc.)

The topics covered include:

1. Fundamental theorem of arithmetic

2. Congruences, Chinese remainder theorem, Euler-Fermat theorem, Lagrange's theorem on solutions to polynomial congruences

3. Quadratic residues mod p, Legendre symbol and Euler's criterion, Gauss' lemma, law of quadratic reciprocity, Jacobi symbol

4. Binary quadratic forms, representation of integers (in particular as sum of two squares)

5. Distribution of the primes, Riemann zeta-function, Dirichlet's theorem, Legendre's formula, Bertrand's postulate

6. Continued fractions, solutions to Pell's equation

7. Primality testing: Fermat, Euler, and strong pseudoprimes

8. Factorisation: Fermat factorisation, factor base methods

These cards contain definitions, statements and proofs of theorems from the course.

I wrote these only to aid with remembering facts for the exam - I've provided them here in the hope that they'll be helpful to someone taking an equivalent course in the same position.

If you aren't taking a similar course, I recommend learning about the material from a textbook rather than attempting to use these cards to teach yourself since I've not included any of the explanation or motivation that accompanied the theorems. I found Davenport's "The Higher Arithmetic" to be an accessible introduction to the subject.

This may go without saying, but a fellow student has had a little trouble along these lines before: if you are taking a course in this subject, these notes are not a substitute for attending lectures.

A note on the cards: I found the LaTeX renderer used by Mnemosyne to be ugly and difficult to read, so the cards are in fact PNG images rendered from LaTeX. This does mean they are a pain to edit, and may not fit nicely on mobile screens. If anyone would like the source files to render themselves, email [email protected] and I will provide them.

## Source:

Based on my lecture notes from the Cambridge University part II course "Number Theory", lectured in Michaelmas term 2016 by Dr. Tom Fisher

Dr. Fisher's website is here, https://www.dpmms.cam.ac.uk/~taf1000/teaching.html and contains example sheets (though no printed notes) for the course.

## Mnemosyne Compatibility:

- 2.1+