A conjectura de Euler sobre somas de potências quárticas de números inteiros
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Data
2017-07-11
Autores
Lopes, Gislayni Telles Vieira Santana
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Universidade Federal do Espírito Santo
Resumo
In 1772, Leonard Euler conjectured that the sum of n powers of positive integers of a given exponent n would also be such a power. However, if the number of powers in this sum is less than the exponent, then such sum could not result in an exponent power n. In the present work we will focus on the n = 4 case of the Euler’s conjecture. In a first approach, we will present a counterexample to the conjecture, that is, we will display positive whole solution for the diophantine equation a 4 +b 4 +c 4 = e 4 , which is equivalent to verify that the set of rational points of the surface S1 : r 4 + s 4 + t 4 = 1 is not empty. We will use the theory of elliptic curves and concepts of Number Theory, such as quadratic reciprocity and Legendre’s theorem, in the construction of a method to obtain the counterexample. In a second approach, we will use the group structure of an elliptic curve to show that there is an infinity of positive integer solutions for the above Diophantine equation if we add a quartic power of an integer in that sum.
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Euler conjecture , Elliptic curves , Diophantine equations , Euler, Conjectura de