Bachet equation

The Bachet equation (English Bachet's equation, after Claude Gaspar Bachet de Meziriac (1581-1638), or Mordell's equation, after Louis Mordell ) is an equation in number theory , which was established in 1650 by Pierre de Fermat and with the last fermatschen Sentence is related.

It is: ${\ displaystyle y ^ {2} -x ^ {3} = c}$

It is interesting how many solutions (integer or rational) are possible for and as a function of . For example , there are only two integer solutions: and , or and . This is also why 26 is the only number that is between a square number and a cube number. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle c}$ ${\ displaystyle c = -2}$ ${\ displaystyle y = 5}$ ${\ displaystyle x = 3}$ ${\ displaystyle y = -5}$ ${\ displaystyle x = 3}$

First of all, Fermat 1650 sets the task of proving that only these two solutions have. However, none of Femat's contemporaries could handle this. Leonhard Euler also tried this problem in 1730. However, his solution was flawed. It was not until 1908 that Axel Thue was able to prove that for every integer other than zero there is only a finite number of integer solutions for and . ${\ displaystyle y ^ {2} -x ^ {3} = - 2}$ ${\ displaystyle y ^ {2} -x ^ {3} = c}$ ${\ displaystyle c}$ ${\ displaystyle x}$ ${\ displaystyle y}$

For there with exactly eight different solutions . ${\ displaystyle c = 17}$ ${\ displaystyle y> 0}$

On the other hand, if one is interested in rational solutions, one can show that if the equation has a possible solution, it automatically has an infinite number of solutions. This was found out by Bachet.

Assuming there is a solution to the equation, there is also a solution to the equation. ${\ displaystyle (x, y)}$ ${\ displaystyle \ left ({\ frac {x ^ {4} -8cx} {4y ^ {2}}}, {\ frac {8c ^ {2} -20cx ^ {3} -x ^ {6}} { 8y ^ {3}}} \ right)}$

This is also known as Bachet's Duplication Formula , which Bachet discovered in 1621.

literature

Introductory text to Rational Points on Elliptic Curves by Joseph H. Silverman and John Tate, Springer-Verlag 1992
U. Felgner: On Bachet's Diophantine equation x ³ = y ² + k . Monthly Math. 98 (3). 1984, 185-191.