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Diophantine equations are a class of polynomial equations for which we seek integer solutions. Named after the ancient Greek mathematician Diophantus, these equations are typically of the form: \[ P(x_1, x_2, ..., x_n) = 0 \] where \( P \) is a polynomial with integer coefficients, and \( x_1, x_2, ..., x_n \) are unknown variables that we want to solve for in the integers.

Integer partitions refer to the ways of expressing a positive integer as the sum of one or more positive integers. The order of terms in each sum does not matter; for example, the two sums \(4 = 1 + 1 + 1 + 1\) and \(4 = 2 + 2\) represent two distinct partitions of the integer 4.

Number theorists are mathematicians who specialize in the field of number theory, which is a branch of pure mathematics focused on the study of the properties and relationships of integers. Number theory encompasses a variety of topics, including: 1. **Prime Numbers**: Study of prime numbers, including their distribution, properties, and related theorems (such as the Prime Number Theorem).

In the context of Wikipedia and other collaborative online encyclopedias, a "stub" is a type of article that is considered incomplete or lacking in detail. A "Number theory stub" specifically refers to a very brief article related to the field of number theory—a branch of pure mathematics devoted to the study of the integers and their properties. Stubs typically provide only basic information or a limited overview of the topic, and they are often marked with a template indicating that they need expansion.

P-adic numbers are a system of numbers introduced by the mathematician Kurt Hensel in 1897, which extends the concept of the usual rational numbers. They are constructed in a way that allows for a different notion of "closeness" between numbers, based on a chosen prime number \( p \). The core idea of p-adic numbers is to define a distance between numbers that is based on divisibility by a prime \( p \).

In number theory, "squares" refers to the squares of whole numbers. A square of a number is the result of multiplying that number by itself. For example, the square of 2 (written as \(2^2\)) is \(2 \times 2 = 4\), and the square of 3 (written as \(3^2\)) is \(3 \times 3 = 9\).

In number theory, theorems are established propositions that are proven to be true based on previously accepted statements, such as axioms and previously proven theorems. Number theory itself is a branch of mathematics that deals with the properties and relationships of numbers, especially integers.

Unsolved problems in number theory are deep questions and conjectures about integers and their properties that have not yet been resolved. Some of the most famous unsolved problems in this field include: 1. **The Riemann Hypothesis**: This conjecture concerns the distribution of the zeros of the Riemann zeta function and has profound implications for the distribution of prime numbers.