Other results

Lagrange's Theorem

Lagrange's theorem states that every positive integer can be represented as a sum of four squares, i.e., $a^2+b^2+c^2+d^2$. For example, the number 123 can be represented as the sum $8^2+5^2+5^2+3^2$.

Zeckendorf's Theorem

Zeckendorf's theorem states that every positive integer has a unique representation as a sum of Fibonacci numbers such that no two numbers are equal or consecutive Fibonacci numbers. For example, the number 74 can be represented as the sum $55+13+5+1$.

Pythagorean triples

A Pythagorean triple is a triple $(a,b,c)$ that satisfies the Pythagorean theorem $a^2+b^2=c^2$, which means that there is a right triangle with side lengths $a$, $b$ and $c$. For example, $(3,4,5)$ is a Pythagorean triple.

If $(a,b,c)$ is a Pythagorean triple, all triples of the form $(ka,kb,kc)$ are also Pythagorean triples where $k>1$. A Pythagorean triple is primitive if $a$, $b$ and $c$ are coprime, and all Pythagorean triples can be constructed from primitive triples using a multiplier $k$.

Euclid's formula can be used to produce all primitive Pythagorean triples. Each such triple is of the form

\[(n^2-m^2,2nm,n^2+m^2),\]

where $0<m<n$, $n$ and $m$ are coprime and at least one of $n$ and $m$ is even. For example, when $m=1$ and $n=2$, the formula produces the smallest Pythagorean triple

\[ (2^2-1^2,2\cdot2\cdot1,2^2+1^2)=(3,4,5) \]

Wilson's Theorem

Wilson's theorem states that a number $n$ is prime exactly when

\[ (n-1)! \bmod n = n-1 \]

For example, the number 11 is prime, because

\[ 10! \bmod 11 = 10 \] and the number 12 is not prime, because

\[ 11! \bmod 12 = 0 \neq 11 \]

Hence, Wilson's theorem can be used to find out whether a number is prime. However, in practice, the theorem cannot be applied to large values of $n$, because it is difficult to calculate values of $(n-1)!$ when $n$ is large.