A proof of Bertrand's postulate

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1

On a Deterministic Property of the Category of kk-almost Primes: A Deterministic Structure Based on a Linear Function for Redefining the kk-almost Primes (∃n∈N\exists n\in {\rm N} , 1≤k≤n1{\le} k {\le}n) in Certain Intervals

In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) $k$-almost primes ($where$ $\exists n\in {\rm N}$, $1{\le} k {\le}n$) in certain interval. A theorem has been proven showing a new deterministic property of the category of $k$-almost primes. Through a linear function that we obtain, an equivalent redefinition of the $k$-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function's property some relations which contain new information about the $k$-almost primes (including primes) are presented.Comment: 10 pages. Accepted and presented article in the 11th ANTS, Korea, 2014. The 11th ANTS is one of international satellite conferences of ICM 2014: The 27th International Congress of Mathematicians, Korea. (Expanded version

Paul Erdös i dokazi iz Knjige

U prvom poglavlju ovog diplomskog rada prikazan je život mađarskog matematičara iz 20. stoljeća, Paula Erdösa. U drugom poglavlju rada nalaze se odabrani dokazi iz triju matematičkih područja, teorije brojeve, kombinatorike i teorije grafova. U Teoriji brojeva iznijet je dokaz tvrdnje da je skup prostih brojeva beskonačan i dokaz Bertrandovog postulata. U Kombinatorici je prikazan dokaz Teorema Erdös-Ko-Rado, a u Teoriji grafova dokaz Turánovog teorema i Teorema o prijateljstvu.In the first chapter of this graduate thesis is presented the life of Hungarian mathematician from the 20th century, Paul Erdös. In the second chapter there are selected proofs from three mathematical areas, number theory, combinatorics and graph theory. Section Number theory gives proof of the infinity of primes and proof of Bertrand's postulate. Combinatorics gives proof of Erdös-Ko-Rado theorem and Graph theory gives the proof of Turán's theorem and Friendship Theorem