Let \(f,g\) be entire functions satisfying \[ f^n+g^n=1 \] for an integer \(n>2\). Prove that \(f\) and \(g\) are constant.
Hint: consider \(f/g\).
Solution
If \(g\equiv0\), then \(f^n\equiv1\), so \(f\) is constant. Assume therefore that \(g\not\equiv0\) and consider the meromorphic function \[ h=\frac{f}{g}:\mathbb C\longrightarrow\widehat{\mathbb C}. \]
1. The function \(h\) omits every \(n\)th root of \(-1\).
Let \(\zeta\in\mathbb C\) satisfy \[ \zeta^n=-1. \] Suppose for contradiction that \[ h(z_0)=\zeta. \] Then \(h(z_0)\) is finite, so \(g(z_0)\ne0\), and \[ f(z_0)=\zeta g(z_0). \] Substituting into Fermat’s equation gives \[ 1 =f(z_0)^n+g(z_0)^n =(\zeta^n+1)g(z_0)^n =0, \] a contradiction. Thus every root of \(z^n=-1\) is omitted by \(h\).
2. The meromorphic function \(h\) is constant.
Because \(n>2\), the equation \[ \zeta^n=-1 \] has at least three distinct roots. By 1, the meromorphic function \(h\) omits all of them. The Little Picard theorem for meromorphic functions states that a nonconstant meromorphic function on \(\mathbb C\) can omit at most two values of the Riemann sphere. Hence \(h\) must be constant, say \[ h\equiv c. \]
3. Both \(f\) and \(g\) are constant.
From \(f=cg\) and \[ f^n+g^n=1 \] we obtain \[ (c^n+1)g^n=1. \] In particular \(c^n+1\ne0\), and therefore \[ g^n\equiv\frac{1}{c^n+1} \] is constant and nonzero. Differentiating gives \[ ng^{n-1}g'=0. \] Since \(g\) never vanishes, \(g'\equiv0\), so \(g\) is constant. Then \(f=cg\) is constant as well.