| Arithmetics and number theory |
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| Teaching Staff in Charge |
| Assoc.Prof. COVACI Rodica, Ph.D., rcovaci@math.ubbcluj.ro Prof. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.ro |
| Aims |
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Presentation of the basic elements of the subject; introduction of number sets, study of divisibility, congruences and arithmetic functions. |
| Content |
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1. Number sets and their structuring: axiomatic study (Peano) of the set N of natural numbers; introducing of the sets Z and Q of integers, respectively rational numbers; operations and order on N, Z and Q; mathematical induction (different forms).
2. Divisibility in N and Z: the divisibility relation in N and in Z, the division algorithm with remainder, representation of natural numbers in a given scale, prime numbers, the fundamental theorem of Arithmetics (the fundamental factorization theorem), the theorem of Euclid on prime numbers, the general division criterion, greatest common divisor, Euclidean algorithm. 3. Congruences and diophantine equations: congruence in Z, the theorem of Euler-Fermat, the theorem of Wilson and the converse, algebraic congruences of first degree, diophantine equations of first degree. 4. Arithmetical functions: Euler's function, number and sum of divisors. |
| References |
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1. Becheanu,M. si colectiv, Algebra pentru perfectionarea profesorilor, Ed. Didactica si Pedagogica, Bucuresti, 1983.
2. Both,N., Aritmetica si teoria numerelor pentru perfectionarea profesorilor, Lito Cluj, 1981. 3. Kulikov,L.J., Algebre et theorie des nombres, Ed. Mir, 1982. 4. Popovici,C., Teoria numerelor, Ed. Didactica si Pedagogica, Bucuresti, 1973. |
| Assessment |
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Exam. |