The thesis discusses classical number theory problems on representations of integers by sums of two, three or four squares. The theorem on representation of prime numbers as sum of two squares, which is presented as first, is proven using different mathematical techniques in 8 different ways (using involutions, pigeon hole principle, convex subsets, integer partitions, method of infinite descent, Gaussian integers, congruences and geometric forms). Theorems on representations of natural numbers by sums of two or by four squares are also proven in two different ways and also some interesting results on the number of such representations are given. Finally, the theorem on representation of natural numbers by sums of three squares is proven using the classical aproach by Dirichlet with quadratic forms and properties of quadratic congruences.
In the conclusion we discuss some possible approaches to communicate these problems in form of math investigations to students in primary or secondary schools. For this purpose we have created a worksheet, which teachers could use to implement such activity in their classroom.
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