Diophantine equations in the context of Algebraic surface

⭐ Core Definition: Diophantine equations

In mathematics, a Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates the sum of two or more unknowns, with coefficients, to a constant. An exponential Diophantine equation is one in which unknowns can appear in exponents.

Diophantine problems have fewer equations than unknowns and involve finding integers that solve all equations simultaneously. Because such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry.

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⭐ Core Definition: Diophantine equations
Diophantine equations in the context of Indeterminate equation
Diophantine equations in the context of Hilbert's eighth problem

Diophantine equations in the context of Indeterminate equation

In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equations are often called Diophantine equations.

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Diophantine equations in the context of Hilbert's eighth problem

Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns various branches of number theory, and is actually a set of three different problems:

the original Riemann hypothesis for the Riemann zeta function
the solvability of two-variable, linear, diophantine equations in prime numbers (where the twin prime conjecture and Goldbach conjecture are special cases of this equation)
the generalization of methods using the Riemann zeta function to estimate distribution of primes in integers to Dedekind zeta functions, and to use them for distribution of prime ideals in a ring of integers of arbitrary number fields.

Along with Hilbert's sixteenth problem, it became one of the hardest problems on the list, with very few particular results towards its solution. After a century, the Riemann hypothesis was listed as one of Smale's problems and the Millennium Prize Problems. The twin prime conjecture and Goldbach conjecture being special cases of linear diophantine equations became two of four Landau problems.

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Diophantine equations in the context of Algebraic surface

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⭐ Core Definition: Diophantine equations

Diophantine equations in the context of Indeterminate equation

Diophantine equations in the context of Hilbert's eighth problem