Diophantine Equations in Algebraic Number Theory

Diophantine equations, classical problems in number theory, seek integer or rational solutions to polynomial relations. When approached through algebraic number theory, these equations are studied over rings of integers in number fields, allowing the deployment of tools such as field extensions, Galois theory and valuation methods. Central themes include Thue and Thue–Mahler equations, where one examines forms of fixed degree, and unit equations, which characterise solutions in multiplicative groups of algebraic integers. The advent of transcendence techniques and explicit height bounds has yielded effective finiteness results, transforming many long-standing existence theorems into algorithms. Interplay with the geometry of numbers further enriches the subject, providing lattice-point methods to refine counts of solutions. These advances have not only resolved classical instances—such as generalisations of the Pell equation and Catalan’s conjecture—but also underlie modern cryptographic protocols and coding theory, where the hardness of certain Diophantine problems ensures practical security. The field continues to expand through refined height inequalities, improved reduction procedures and deeper insights into the arithmetic of elliptic and higher-genus curves.

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Diophantine Equations in Algebraic Number Theory publication trend

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Technical terms

Diophantine equation: A polynomial equation for which integer or rational solutions are sought.

Algebraic number field: A finite extension of the rational numbers, whose elements satisfy polynomial equations with rational coefficients.

Unit equation: A relation of the form x + y = 1 where x and y belong to the multiplicative group of units in a ring of algebraic integers.

Height function: A measure of arithmetic complexity of algebraic numbers, used to bound and count solutions.

Baker’s method: A transcendence approach providing explicit lower bounds for linear forms in logarithms of algebraic numbers, crucial for effective Diophantine results.

References

1. On the x–coordinates of Pell equations which are k–generalized Fibonacci numbers. Journal of Number Theory (2020).
2. Repdigits as Product of Terms of k-Bonacci Sequences. Mathematics (2021).
3. S-unit equation in two variables and Padé approximations. International Journal of Number Theory (2023).