• On the Diophantine equation 2...

Treffer: On the Diophantine equation 2 + p = m2 with a Fermat prime p: On the Diophantine equation \(2^s + p^k = m^2\) with a Fermat prime \(p\)

Title:
On the Diophantine equation 2 + p = m2 with a Fermat prime p: On the Diophantine equation \(2^s + p^k = m^2\) with a Fermat prime \(p\)
Authors:
Luca, Florian, Pink, István
Source:
Journal of Number Theory. 268:49-71
Publisher Information:
Elsevier BV, 2025.
Publication Year:
2025
Subject Terms:
\(p\)-adic approximation lattices, linear form in logarithms, continued fractions, Exponential Diophantine equations, Higher degree equations, Fermat's equation, Linear forms in logarithms, Baker's method
Document Type:
Fachzeitschrift Article
File Description:
application/xml
Language:
English
ISSN:
0022-314X
DOI:
10.1016/j.jnt.2024.09.006
Access URL:
https://zbmath.org/7956622
https://doi.org/10.1016/j.jnt.2024.09.006
Rights:
Elsevier TDM
Accession Number:
edsair.doi.dedup.....6ccd1e4aa020adda3b5c53291b80f6f5
Database:
OpenAIRE

Weitere Informationen

The authors find all the non-negative integer solutions \((m,p, k,s)\) of the equation \(2^s+p^k=m^2\). The set of solutions has 4 tuples that are finite and three with \(k=0, 1\) that come with a parameter each. The methods include considerations in the ring of integers of \(\mathbb{Q}(\sqrt{2})\), an application of the Carmichael's primitive divisor theorem, lower bounds for linear forms in logarithms, and computations using MAGMA.