Diophantine Number Theory
A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".
Deadline for manuscript submissions: closed (31 October 2021) | Viewed by 3705
Special Issue Editor
Interests: number theory; diophantine equations; polynomials
Special Issues, Collections and Topics in MDPI journals
Special Issue Information
Dear colleagues,
Number theory and especially Diophantine equations are the most classical topics of mathematics. For example, one can think of Pythagorean triplets. Somehow, these ancient objects show that Diophantine equations also useful for other topics of mathematics. Some of the fundamental questions to handle these equations are how to give an effective or ineffective finiteness result for the number of solutions, how to give an effective or ineffective finiteness theorem for the size of solutions, and finally, how to resolve the equations. The last problem is sometimes extremely hard—see, for example, the Fermat Last Theorem (FLT) or the Catalan problem. There is no general algorithm that can resolve an arbitrary Diophantine problem, so certain special classes of equations, including two-variables equations (S-unit, Thue, and super-elliptic equations) and multivariable equations (decomposable form, discriminant, and norm form equations), are very important. The theory of the previous families is well-known. Unfortunately, there are only a few general methods for solving them. The first one is Baker theory. Roughly speaking, this method states that a linear form of logarithms is zero or its absolute value is greater than a positive, effectively computable constant. Here, we must mention the subspace theorem for multivariable equations. The second method is also the most powerful one, i.e., the so-called modular technique, which is what Andrew Wiles used to prove the classical conjecture by Fermat (FLT).
I believe that the most fruitful approach is to combine Baker theory and the modular technique. We cannot list all the applications of these profound methods, of course, but this thematic issue will provide a valuable source of high-quality research papers and surveys on the modern theory of Diophantine equations. The Diophantine properties of polynomials and recurrence sequences will also be considered. Usually, the Diophantine equation means an equation over the integer number, but we shall study certain general Diophantine equations over algebraic integers and over polynomials (Mason’s inequality).
Prof. Dr. Ákos Pintér
Guest Editor
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Keywords
- Diophantine equations
- symmetric Diophantine equations
- unit equations
- polynomial and exponential Diophantine equations
- Fermat last theorem
- modular method
