A Small Contribution to Catalan′s Equation

✍ Scribed by A.M.W. Glass; D.B. Meronk; T. Okada; R.P. Steiner

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 222 KB
• Volume: 47
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1994.1030

✦ Synopsis

Using recent results on linear forms in logarithms of algebraic numbers, we prove that any solution of the equation (x^{p}-y^{q}=\varepsilon), where (\varepsilon= \pm 1, p) and (q) are odd primes, and (p>q) satisfies (p<3.42 \cdot 10^{28}) and (q<5.6 \cdot 10^{19}). We also combine our work with some results of Altonen and Inkeri to determine the six cases with (q \leqslant 37) for which this equation may have solutions. 1994 Academic Press. Inc.