has no solutions in positive integers
We know (see example of Fermat’s Last Theorem) that the sum of two fourth powers can never be a square unless all are zero. This article shows that the difference of two fourth powers can never be a square unless at least one of the numbers is zero. Fermat proved this fact as part of his proof that the area of a right triangle with integral sides is never a square; see the corollary below. The proof of the main theorem is a great example of the method of infinite descent.
has no solutions in positive integers.
Suppose the equation has a solution in positive integers, and choose a solution that minimizes . Note that , and are pairwise coprime, since otherwise we could divide out by their common divisor to get a smaller solution. Thus
Factoring the original equation, we get
If , then , and clearly , so we have found a solution smaller than the assumed minimal solution.
Assume therefore that . Now, ; we may assume by relabeling if necessary that is even and odd. Then are pairwise coprime and form a pythagorean triple; thus there are of opposite parity and coprime such that
is a square; it follows that , and are all (nonzero) squares since they are pairwise coprime. Write
for positive integers . Then , and
We have thus found a smaller solution in positive integers, contradicting the hypothesis. ∎
No right triangle with integral sides has area that is an integral square.
Suppose is a right triangle with the hypotenuse, and let . Either or is even; by relabeling if necessary, assume is even. Then we can choose relatively prime integers with and of opposite parity such that
If the triangle’s area is to be a square, then
must be a square, and thus must be a square. Since and are coprime, it follows that , , and are all squares, and thus that is the difference of two fourth powers. But then
must also be a square. Since both and are squares, this is impossible by the theorem. ∎
|Title||has no solutions in positive integers|
|Date of creation||2013-03-22 17:05:04|
|Last modified on||2013-03-22 17:05:04|
|Last modified by||rm50 (10146)|