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Every integer greater than 1 can be uniquely factorized into prime numbers. This theorem ensures that we can break down any number into a product of its prime factors in only one way, excluding the rearrangement of the prime factors.

A Pythagorean triple consists of three positive integers $a$, $b$, and $c$, such that $a^2 + b^2 = c^2$. These are the integer solutions to the Pythagorean theorem for right triangles.

Legendre's Conjecture suggests that there is at least one prime number between any two consecutive squares $n^2$ and $(n+1)^2$. This conjecture has not been proven or disproven.

The theorem states that every natural number can be represented as the sum of four integer squares. That is, for every $n \geq 0$, there exist integers $x, y, z, t$ such that $n = x^2 + y^2 + z^2 + t^2$.

Euler's Criterion is a statement in number theory that gives a condition for determining whether an integer is a quadratic residue modulo a prime. For an integer $a$ and an odd prime $p$, $a$ is a quadratic residue modulo $p$ if and only if $a^{\frac{p-1}{2}} \equiv 1 \pmod p$.

Euler's Theorem is a generalization of Fermat's Little Theorem. It states that if $n$ is a positive integer and $a$ is an integer coprime to $n$, then $a^{\varphi(n)} \equiv 1 \pmod n$ where $\varphi$ is Euler's totient function.

The theorem states that for any given two positive coprime integers $a$ and $d$, there are infinitely many primes of the form $a + nd$, where $n$ is a non-negative integer. This result shows that there are numerous numbers with the same difference between them that are prime.

Carmichael's Theorem is a result in number theory that generalizes Fermat's little theorem. It states that for Carmichael numbers $n$, for all integers $a$ that are coprime to $n$, it holds that $a^{n-1} \equiv 1 \pmod n$.

The Density of Primes, given by the Prime Number Theorem, approximates the distribution of the primes by stating that the probability of a number around $n$ being prime is about $\frac{1}{\ln(n)}$, where $\ln$ is the natural logarithm.

If a prime $p$ divides the product $ab$ of two integers $a$ and $b$, then $p$ must divide at least one of those integers $a$ or $b$. This is fundamental in the study of number theory because it leads to the unique factorization of integers.

If $p$ is a prime number, then for any integer $a$, the number $a^p - a$ is an integer multiple of $p$. In the notation of modular arithmetic, this is expressed as $a^p \equiv a \pmod p$.

Bezout's Identity states that for any non-zero integers $a$ and $b$, there exist integers $x$ and $y$ such that $ax + by = \gcd(a, b)$, where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$.

The Goldbach Conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even integer greater than 2 is the sum of two prime numbers.

In number theory, Gauss's Lemma relates to the Legendre symbol and gives a way to compute it. For a given odd prime $p$ and an integer $a$ not divisible by $p$, the Legendre symbol $(\frac{a}{p})$ is equal to $(-1)^n$ where $n$ is the number of integers $k$ such that $\frac{p}{2} < a k \mod p < p$.

The Möbius function $\mu(n)$ is a multiplicative function that is used in number theory and combinatorics. It is defined for a positive integer $n$ such that:

The Pell Equation is a diophantine equation of the form $x^2 - ny^2 = 1$ where $n$ is a non-square natural number. The solutions $(x, y)$ are called Pell numbers. This equation is central to the theory of quadratic forms and has an infinite number of solutions.

In number theory, the Partition Function $P(n)$ represents the number of distinct ways of representing $n$ as the sum of positive integers, without considering the order of addends. For example, $P(4) = 5$ because 4 can be partitioned as 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.

Fermat's Last Theorem states that there are no three positive integers $a, b,$ and $c$ such that $a^n + b^n = c^n$ for any integer value of $n$ greater than 2. This was famously conjectured by Pierre de Fermat in 1637 and was only proven in 1994 by Andrew Wiles.

An integer $p > 1$ is a prime number if and only if $(p - 1)! + 1$ is divisible by $p$ where $!$ denotes the factorial operation. In modular arithmetic, this can be written as $(p - 1)! \equiv -1 \pmod p$.

Euler's Totient Function $\varphi(n)$ counts the number of positive integers less than or equal to $n$ that are coprime to $n$. It is a key function in number theory, particularly in modular arithmetic and theorems involving congruences.

Given two integers $a$ and $b$, with $b > 0$, there exists unique integers $q$ and $r$ such that $a = bq + r$ and $0 \leq r < b$. The number $q$ is called the quotient, while $r$ is the remainder.

This is a theorem in number theory that allows the determination of the solvability of quadratic equations modulo prime numbers. It provides a relationship between the Legendre symbols $(\frac{p}{q})$ and $(\frac{q}{p})$ for any two odd primes $p$ and $q$.

A Sophie Germain prime is a prime number $p$ such that $2p+1$ is also prime. The number $2p+1$ is called a safe prime. These primes are named after French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem for the case $n=5$.

A Mersenne prime is a prime number of the form $M_n = 2^n - 1$ for some integer $n$. Not all numbers of this form are prime, but primes of this form are used in large number cryptography.

The Twin Prime Conjecture asserts that there are infinitely many pairs of prime numbers (p, p+2) such that both numbers are prime. A pair of primes is called a 'twin prime'.

The Riemann Hypothesis is a conjecture that the nontrivial zeroes of the Riemann zeta function all have real part 1/2. This hypothesis has profound implications on the distribution of prime numbers.

The Pigeonhole Principle states that if $n$ items are put into $m$ containers, with $n > m$, then at least one container must contain more than one item. This simple yet powerful principle is used in various mathematical proofs.

If one knows the remainders of the division of an integer $n$ by several pairwise coprime integers, then one can determine uniquely the remainder of the division of $n$ by the product of these integers, up to the product of these integers.