Sequences of powers with second differences equal to two and hyperbolicity. (English) Zbl 1440.11110
Summary: By explicitly finding the complete set of curves of genus 0 or 1 in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound \(M>0\) such that there are only finitely many sequences of length \( M\) formed by \(k\)-th rational powers with second differences equal to 2. Moreover, we prove the unconditional analogue of this result for function fields, with \(M\) depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11D41 | Higher degree equations; Fermat’s equation |
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 14G05 | Rational points |
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