How can we describe the solutions in positive integers of x² + y²  = z² ?  This is “classical'' (there is a connection with Pythagoras' Theorem). Analogous questions include the solution of x² + y²  = Kz², with K = 2 or 3, for example. When K = 3 there are no solutions: this can be shown using congruence considerations and Fermat's notion of “infinite descent'', which is just mathematical induction expressed in a different way.

A different but related question is: which integers n (not necessarily squares) are representable as x² + y² , and, if they are, then in how many ways? An illuminating way of considering this question is via the study of Gaussian Integers x + iy, where i² = - 1. This involves the study of Gaussian Primes and the uniqueness of factorisation of Gaussian Integers as products of Gaussian Primes.

Fermat's “Little'' Theorem (nothing to do with “Fermat's Last Theorem'') says that a^p -a is divisible by p when p is prime. Historically, this result seems to have arisen from questions involving “perfect'' numbers n (whose factors sum to 2n), but the result has far greater significance elsewhere. It is needed, for example, in a characterisation of those primes p which divide numbers of the form n²+1 (they are precisely those p which are not of the form 4k+3), a fact which is indispensable for a study of Gaussian Primes and Sums of Two Squares.

Prerequisite Modules: MA0111 Elementary Number Theory I

• Formulate problems in terms of solutions of systems of equations in integers.
• Solve systems of linear equations in integers.
• Solve simple examples of non-linear Diophantine equations.
• Prove that suitable Diophantine equations have no solutions by congruence considerations and the method of infinite descent.
• Compute the order of a given number to a given modulus
• Use Fermat's Theorem to characterise the primes that divide a suitable polynomial expression.
• Factorise Gaussian Integers as products of Gaussian Primes.
• In simple cases, determine how many representations a given integer has as a sum of two Squares.
• Determine good rational approximations to given real numbers, using the Continued Fraction Algorithm.
• Find solutions to instances of Pell's Equation using the Continued Fraction Algorithm.
• Understand the proofs of the theorems underlying the procedures described in the course.

27 - 50 minute lectures

Some handouts will be provided in hard copy or on a website, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

Problem formulation. Solution of simple Diophantine Equations. Working with Gaussian Integers and with Continued Fractions.

Transferable Skills:

Development of mathematical tools to solve problems of a type not previously encountered: assessing a problem and relating it to previous examples, or generalising previous examples to construct a new argument.

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Revision of the Fundamental Theorem of Arithmetic and its implications for Linear Diophantine Equations and Congruences.
• Simple examples of non-linear congruences. Fermat's “Method of Descent'' applied via congruence considerations to examples of Diophantine Equations having no solutions.
• The equation x² + y²  = z² .
• The Fermat--Euler Theorem. Applications to prime divisors of the values of suitable polynomial expressions.
• Gaussian Integers: uniqueness of factorisation. Characterisation of Gaussian Primes. Application to the representability of integers as sums of two squares of rational integers.
• Expression of rationals and quadratic irrationals as Simple Continued Fractions. Properties of the convergents. Pell's Equation: properties of the solutions, and their calculation from the convergents of the appropriate continued fraction.

Please see Background Reading List for an indicative list.

Elementary Number Theory, Jones, G. A., & Jones, J. M., Springer

Elementary Number Theory and its Applications, Rosen, K. H., Addison-Wesley

The Higher Arithmetic, Davenport, H, Cambridge