Number theory is one of branches of mathematics which deals with relations between Integer Numbers (positive and negative natural numbers and zero) which shown as Z numbers.
Number theory, as a course in undergraduate of mathematics, usually starts with topic of “Divisibility”. This section involves with some sub-topics like “division algorithm”, “Greatest Common Divisor”, “Least Common Multiple”, “Euclid Algorithm”, “Prime Numbers”, “Sieve of Eratosthenes”, “Fundamental Theorem of Arithmetic”, “Linear Diophantine Equations” and etc.
There are some famous theorems and conjectures related to this section which are: “Chebyshev’s Theorem”, “Dirichlet’s Theorem”, “Bertrand’s conjecture” and “Goldbach’s conjecture”.
Another section of this course is “Modular Arithmetic” which is in concept is like “Divisibility” but it has its own terminology and its own theorems to solve number theory problems. This section involves with some sub-topics like: “Residue Numeral System (RNS)”, “Modular Equations System”, “Primitive Root Modulo of n”, “Chinese Reminder Theorem” and etc.
There are some theorems in this section too which the most famous one that usually taught in universities are: “Fermat’s little theorem”, “Hensel’s Lemma”, “Euler’s Totient Function”, “Euler’s Theorem” and “Wilson’s Theorem”.
In summary, “Number Theory” is almost a branch of “Pure Mathematics” which means, all of the theorems and applications only involves in mathematics and there is no application of it in other sciences or in a real world scenario. Although, there are some applications of “Number Theory” in “Cryptography” and Encryption / Decryption methods and maybe by growing other fields like this, we observe more application of number theory in other sciences. It is also worth mentioning that this field of Mathematics and its problems are very hard to study and as you will see when involving with this field, there are many unsolved problems and conjectures in “Number Theory”.