Divisibility. Congruences. Euler’s function. Euler’s theorem. Primitive roots. Simple Diophantine equations. Quadratic reciprocity. Primality tests. The Lucas-Lehmer criterion. Continuous fractions and Diophantine approximations. Comments on the Lagrange spectrum. Khintchine’s theorem. Asymptotic estimates of arithmetic functions. Dirichlet’s theorem. The prime number theorem.

References:
HARDY, G. H., WRIGHT, E. M. – An introduction to the theory of numbers, 3rd ed., Oxford, at the Clarendon Press, 1954.
IRELAND, K., ROSEN, M. – A classical introduction to modern numbers theory, 2nd ed, New York, Springer-Verlag, 1982 – 1990.
CASSELS, J. W. S. – An introduction to diophantine approximations, Cambridge, at the University Press, 1957.
VINOGRADOV, I. M. – Elements of number theory, Dover, 1954.
MOREIRA, C. G., SALDANHA, N. – Mersenne’s cousins and other very large cousins – 22º Colóquio Brasileiro de Matemática. Rio de Janeiro. Third edition, IMPA, 2008.
BROCHERO, F., MOREIRA, C.G., SALDANHA, N., TENGAN, E. – Number theory – a tour around the world with primes and other familiar numbers, Projeto Euclides, IMPA, 2010.

* Basic syllabus. The teacher has the autonomy to make any changes.