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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Solutions to Some Advanced Methods in Solving Diophantine Equations
verfasst von : Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
Erschienen in: An Introduction to Diophantine Equations
Verlag: Birkhäuser Boston
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1.
Solve the equation
$$x^2 + 4 = y^n,$$
where n is an integer greater than 1
.
Solution
. For
n
= 2, the only solutions are (0, 2) and (0, –2). For
n
= 3, we have seen in Example 4 that the solutions are (2, 2), (–2, 2), (11, 5), and (–11,5). Lef now
n
≥ 4. Clearly, for
n
even, the equation is not solvable, since no other squares differ by 4. For
n
odd, we may assume without loss of generality that
n
is a prime
p
≥ 5. Indeed, if
n
=
q
k
, where
q
is an odd prime, we obtain an equation of the same type:
x
2
+ 4 = (
y
k
)
q
.