Programme

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

Core Documentation

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense
D.M. Burton, Elementary Number theory, McGraw-Hill, (sesta edizione 2007)
G.A. Jones e J.M. Jones, Elementary Number theory, Springer, 1998 (ristampa 2008)
H. Davenport, Aritmetica superiore. Una introduzione alla teoria dei numeri, Zanichelli, (1994)
G.H. Hardy e E.M. Wright, An introduction to the theory of numbers, Oxford University Press, (1979)
K.H. Rosen, Elementary number theory and its applications, Addison-Wesley (2000)

Type of delivery of the course

Lectures, exercises and tutoring are provided

Type of evaluation

The exam will consist of a written and an oral test at the end of the course. During the course there will be two partial tests that will be evaluated as a written exam. To those who pass both tests during the course with a vote higher than 18/30 (for each test) the teacher will propose a vote to verbalize the exam without the need to take an oral test. This proposal may also be refused by students if they wish to take an oral test to try to improve the final result. The oral is however necessary for those who want to aspire to praise. The written test (including the partial tests) consists of 5/6 practical / theoretical exercises to be carried out in 2.30 / 3 hours.