Bezout's Identity [ax+by=d]

In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation. It states that if a and b are nonzero integers with greatest common divisor d, then there exist infinitely many integers x and y (called Bézout numbers or Bézout coefficients) such that

${\displaystyle ax+by=d.\,}$

Additionally, d is the least positive integer for which there are integer solutions x and y for the preceding equation.
The Bézout numbers x and y as above can be determined with the Extended Euclidean algorithm. However, they are not unique. If one solution is given by (x, y), then there are infinitely many solutions. These are given by

${\displaystyle \left\{\left(x+{\frac {kb}{\gcd(a,b)}},\ y-{\frac {ka}{\gcd(a,b)}}\right)\mid k\in \mathbb {Z} \right\}.}$

Extended Euclidean Algorithm

The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of integers a and b: it also finds the integers x and y in Bézout's identity

${\displaystyle ax+by=\gcd(a,b).\,}$

(Typically either x or y is negative). The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the modular multiplicative inverse of a modulo b.

Continued Fractions

Pythagorean Triples

Pythagorean triplets are sets of three natural numbers that satisfy Pythagoras' theorem;${\displaystyle x^{2}+y^{2}=z^{2}}$ which describes the relationships between the sides of a right angled triangle. It has been shown that there are an infinite number of them by the following proof.
The square numbers; 1, 4, 9, 16 etc can be seen to be separated by the odd numbers 3, 5, 7 etc. This is because ${\displaystyle (n+1)^{2}=n^{2}+2n+1}$ As an infinite number of these odd numbers are squares ( as an odd number squared results in an odd number) there must be an infinite number of pythagorean triplets.

Fermat's Last Theorem

Fermats last theorem is a theorem about an equation that is similar to pythagorus' theorem. It is ${\displaystyle x^{n}+y^{n}=z^{n}}$. Fermats last theorem states that there are no integer solutions to this equation for x,y,z does not equal 0 and ${\displaystyle n>2}$. It is particularly famous because Fermat stated that he had a proof. The first case of this to be proved was n=4 which was proved by infinite descent.

Pythagorean Quadruples and higher n-tuples

Chinese Remainder Theorem