has trivial solutions on the line . For other solutions one has the
. The only rational solutions of (1) are
. Consequently, the only integer solution of (1) is
Proof. . Let be a solution of (1) with . Set (). Now
from which we obtain easily
where . Then
. The unit fractions yield from (2) rational solutions (3). Further, no irrational value of cannot make both and of (2) rational, since otherwise the ratio of the latter numbers would be irrational (cf. rational and irrational). Accordingly, for other rational solutions than (3), we must consider the values
has a solution . Thus we may write and rewrite the rational number
This product form tells that is rational. But the number
i.e. could not be a power. The contradictory situation means, by (4), that the antithesis is wrong. Therefore, the numbers (3) give the only rational solutions of (1).
Note. The value in (3) produces , , whence (1) reads
The truth of the equality (5) may also be checked by the calculation
- 1 P. Hohler & P. Gebauer: Kann man ohne Rechner entscheiden, ob oder grösser ist? Elemente der Mathematik 36 (1981).
- 2 J. Sondow & D. Marques: Algebraic and transcendental solutions of some exponential equations. Annales Mathematicae et Informaticae 37 (2010); available directly at http://arxiv.org/pdf/1108.6096.pdfarXiv.
|Date of creation||2014-12-16 16:29:29|
|Last modified on||2014-12-16 16:29:29|
|Last modified by||pahio (2872)|