Here is another one. Prove that the power of $13$ can be writen as a sum of two squares.
I will give two proofs of it. First one is more involved and includes lemma $$(a^2+b^2)(x^2+y^2)= (ax+by)^2+(bx-ay)^2$$ yet second takes step 2 and it is much more elegant.
Base: $n=1$, then $13 = 2^2+3^2$ and we are done.
We know that $13^n = a^2+b^2$ ...
A simple consequnce of:
Postage Stamp Problem, which states that for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$,
is that every natural number greater or equal to $mn-m-n+1$ can be writen in a form $am+bn$ for some $a,b$.
And for some ...
Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\mathbb N$ be a strictly increasing function such that $f(2)=2$ and $f(m)f(n)=f(mn)$ for all positive integers $m,n$ such that $\gcd(m,n)=1$. Find all such functions $f$.
Watered down version:
Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\...
Hmm, I'm surprised than no one mentioned de Moivre's formula: $$(\cos \phi +i\cdot \sin \phi)^n= \cos (n\phi) + i\cdot\sin (n\phi)$$
It is an excellent theorem to prove where we must also use the addition theorem for sin and cos.
Prove that, for all x in N, we have x+1 =/= x. (Requires use of proof by contradiction.)
Use only the following properties of addition on N:
Addition is closed on N.
x+1 =/= 1 for all x in N.
If x+1 = y+1, then x=y for all x, y in N.
The Principle of Mathematical Induction.
1+1 =/= 1 follows directly from (2).