2 Odd Integers Add To Even Integer

Conjecture

$\forall x,y\in O⟹x+y\in E$

Proof

Assumed Axioms

• Arithmetic
• Rules of Algebra

1. An even integer is an integer that can be written as $2n,\forall n\in Z$
2. An odd integer is an integer that can be written as $2n+1,\forall n\in Z$

Proof

Take 2 odd integers $p$ and $q$. By definition, we can write that $p=2n+1$ and $q=2m+1$ for some integers $n\in Z$ and $m\in Z$. Then: $p+q=2n+1+2m+1$

$$=2n+2m+2$$ $$=2(n+m+1)$$

Note that $n+m+1$ is an integer. by definition, $2(n+m+1)$ is an even number. QED