Introduction to Number Theory


The student is supposed to learn the basic concepts of elementary number theory.

General characterization





Responsible teacher

Joaquim Eurico Anes Duarte Nogueira


Weekly - 5

Total - 70

Teaching language





Burton, David, Elementary Number Theory, 2nd ed., Wm. C. Brown, Dubuque, IA, 1989.

Tattersall, James - Elementary Number Theory in Nine Chapters, 2th edition, Cambridge University Press, 2005.

Ore, Oysten., An Invitation to Number Theory, Mathematical Association of America, Washington, DC, 1967.

Sierpinski, W., Elementary Theory of Numbers, North-Holland, Amsterdam, 1988.

G.H.Hardy, E.M.Wright, Theory of numbers.

Teaching method

There are classes in which theory is lectured and illustrated by examples. There are also problem-solving sessions. Some exercises are left to the students to be solved on their own as part of their learning process.

Evaluation method

Evaluation Rules

1. Attendance

To have attendance students will have to attend at least 2/3 of the classes.

2. Continuous assessment

There will be two tests during the semester.

a) All students enrolled in the subject who, at the time of the test, are eligible to take the test or are exempt from taking it, may take it.

b) To obtain the classification of the tests (CT), the student must have obtained a classification of at least 7.5 in the second test.

c) The classification of the tests is obtained by taking the arithmetic mean of the unrounded grades obtained in the three tests, provided that the third test is higher than or equal to 7.5. If the CT (rounded to the nearest unit) is less than 10, the student can take the exam. 
If the CT (rounded up to the nearest whole number) is greater than, or equal to, 10 and less than 18, the student passes with that grade. 
If TC (rounded to the nearest whole number) is greater than, or equal to, 18 the student may choose to take a final grade of 17 or take a supplementary test to defend the grade.

4. Exam

a) Any student who has not yet passed the subject and who has taken or is exempt from taking the examination may sit for the supplementary examination (season 2).

b) If the classification, rounded to the units, is less than 10, the student will fail. 
 If the mark, rounded up to the nearest whole number, is greater than, or equal to, 10 and less than 18, the student passes with that mark. If the grade, rounded up to the nearest whole number, is 18 or more, the student can choose between a final grade of 17 or an additional test to defend the grade.

c) On the exam date the student may choose between taking the exam on the whole subject or taking (or repeating) the part corresponding to one of the tests. In this second situation the rules of continuous assessment will apply.

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Subject matter

  • Divisibility
  • Prime numbers
  • Euclidean algorithm
  • Fundamental thorem of arithmetic
  • Factorization methods
  • Diophantine equations. Pythagorean triples
  • Congruences and modular arithmetic
  • Linear modular equations. Polynomial equations.
  • Chinese remainder theorem
  • Wilson´s theorem and Fermat´s theorem
  • Euler´s theorem and function
  • Perfect numbers
  • Quadratic reciprocity
  • Continued fractions
  • Arithmetic functions
  • Open problems


Programs where the course is taught: