Equations in prime powers
There are some number theoretical questions as Hauptidealsatz, Kronecker classes or arithmetically equivalent fields [c.f. the second author and \textit{D. B. Wales}, J. Algebra 96, 94-113 (1985; Zbl 0577.20005)] where problems of algebraic number theory are translated into group theoretical properties. So it is not surprising that scholars of group theory attempt to investigate number theoretical problems. The authors consider the diophantine equation (*) \((q^ n-1)/(q-1)=p^ a\) with positive integers \(a\), \(q\), \(n\), \(q\geq 2\), \(n\geq 3\), and \(p\) prime. Applying Zsigmondy's theorem on primitive prime divisors it is an exercise to see that \(n\) is prime and if \(q=r^ m\), then \(m\) is a power of \(n\), \(p\equiv 1\pmod{mn}\) [c.f. the reviewer's thesis, Diophantische Probleme bei Kreisteilungspolynomen, FU Berlin (1975), p. 40]. Theorem 1 shows that \((n,a)=1\) if \(r\) is a prime. The other number theoretical results are mainly known (Nagell, Ljunggren, Schinzel) or erroneous, e.g. the class number of \({\mathbb{Q}}(\sqrt{-341})\) is 28 [\textit{Z. I. Borevich} and \textit{I. R. Shafarevich}, Zahlentheorie (1964; Zbl 0121.042), Tabelle 4], so the requirements of corollary 2 are fulfilled, but the equation \(y^ 5=x^ 2+341\) has the solution \(y=377\), \(x=2.759.646\) [c.f. \textit{J. Blass}, Math. Comput. 30, 638-640 (1976; Zbl 0336.10012)]. The error seems to be in Theorem 4 where 331 perhaps should be 341. \{Reviewer's remark: According to the table in Borevich-Shafarevich there would be another counterexample: \(y^ 5=x^ 2+485\) has the solution \(y=9\), \(x=242\). But here the table is wrong: The class number of \({\mathbb{Q}}(\sqrt{-485})\) is 20, not 16 as indicated. -- Friendly communicated by J. Antoniadis.\} Finally the authors sketch a short proof that for a given prime p there are only finitely many a, q, n, \(n\geq 3\), with (*) [c.f. the more general result of \textit{T. Shorey} and \textit{R. Tijdeman}, Math. Scand. 39, 5-18 (1976; Zbl 0341.10017)]. From this they deduce corollary 3: If \(p\) is an odd prime, there are only finitely many simple groups which have a \(p\)-complement (i.e., a subgroup of order prime to \(p\) and of index a power of \(p\)).
scientific article; zbMATH DE number 3929253
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