Representation of Positive Integers as a Sum of Powers of Integers

Zapisivanjem prirodnih brojeva u obliku sume nenegativnih potencija bavili su se mnogi matematičari, a neki iskazani rezultati nisu u potpunosti dokazani ni danas. U sklopu razmatranja koji se prirodni brojevi mogu zapisati u obliku sume kvadrata dva cijela broja okarakterizirani su prosti brojevi s navedenim prikazom. Točnije, prost broj će biti suma kvadrata dva cijela broja ako i samo ako je jednak broju 2 ili ako je oblika 4k+1, k ∈ N. Generalizirajući prethodnu tvrdnju na prirodne brojeve dolazimo do Fermat-Eulerovog teorema koji govori da se jedino prirodni brojevi koji sadrže proste faktore oblika 4k + 3, k ∈ Z, s parnim eksponentom mogu zapisati kao suma kvadrata dva cijela broja. Dana je i formula za određivanje broja takvih prikaza. Razmatrani su i problemi zapisivanja prirodnih brojeva u obliku sume kvadrata dva prirodna broja, te dva relativno prosta prirodna broja. Zatim je naveden Gaussov rezultat vezan za prirodne brojeve koji se mogu prikazati u obliku sume kvadrata tri cijela broja, tj. brojeve koji nisu oblika \(4^{l}(8k+7), k,l\geq 0\). Zanimljiva tvrdnja koja se pripisuje Lagrangeu i Euleru govori da se svaki prirodan broj može zapisati u obliku sume kvadrata četiri cijela broja, a navedeno je i koji se prirodni brojevi mogu prikazati kao suma kvadrata četiri prirodna broja. Iako ne postoji postupak za određivanje prikaza prirodnih brojeva u obliku sume kvadrata pet ili više prirodnih brojeva, navedeni su rezultati koji govore koji se brojevi ne mogu zapisati u tome obliku. Zatim se fokus prebacuje na prikaze u obliku sume kubova cijelih brojeva, a na kraju na završno poopćenje, tj. Waringov problem.Many mathematicians considered the problem of representation of positive integers as a sum of non-negative powers and still some of the results have not been proved. By observing positive integers that can be represented in the form of two squares of integers, we characterized primes in the such form. More precisely, prime number is a sum of two squares of positive integers if and only if it is equal two or it has the form of 4k + 1, k ∈ N. Generalizing the previous statement to positive integers, we have Fermat-Euler’s theorem which states that only positive integers which contain primes of the form 4k+3, k ∈ Z, with the even exponent are the sum of two squares of integers. We also gave the formula for the number of such representations. Moreover, we considered the problem of representation of positive integers as the sum of squares of two positive integers and two coprime positive integers. Furthermore, we stated the result of Gauss concerning the representation of positive integers as the sum of three squares of integers, i.e. numbers which are not of the form \(4^{l}(8k+7), k,l\geq 0\). The interesting statement attributed to Lagrange and Euler states that every positive integer can be written as the sum of four squares of integers. Also, we expressed the positive integers that can be represented as the sum of four squares of positive integers. While there is no algorithm for the representation of positive integers as the sum of five or more squares of positive integers, we mentioned the results concerning to numbers that cannot be represented in that form. Finally, we focused on representation of positive integers as the sum of cubes of integers and considered the generalization known as Waring’s problem