La conjetura de Erdős–Straus

Résumé

Paul Erdös and Ernst G. Straus conjectured in the late 1940s: Given a natural number n greather or equal to 2 it is always possible to decompose the fraction 4/n as sum of three positive rational number with numerator equal to 1 (Egyptian fractions). This conjecture (ESC) is open today. In this paper we study ESC, we establish some conjectures that offer sufficient conditions for the validity of ESC, we give an algorithm which, if it stops, breaks down the fraction 4/n as a sum of three Egyptian fractions, and, for example, we show that ESC holds for all the values of n in the range of a polynomial p(a,b,c) in three variables, linear as function of each variable. We conjecture that the values this polynomial include all the prime numbers congruent to 1 modulus 4, and we have done a computer-assisted verification of this fact for numbers up to 1.2 thousand trillion. On the one hand, we prove that the perfect squares do not belong to the image set of p(a,b,c) but, on the other, with the help of that polynomial we have been able to give a constructive proof that there are arbitrarily long sequences of consecutive numbers for which ESC is true.