# Inductive Proof (+linear equation in four variables)

1. Sep 21, 2011

### sweetreason

I'm trying to prove by induction that $$\forall n \geq 5, \exists m_1, m_2 \in \mathbb{N}$$ such that $$n = 2m_1 +3m_2.$$I've done the base case, and the inductive step boils down to showing that $$\exists m_1 \prime m_2 \prime$$ such that $$2m_1 +3m_2 +1 = 2m_1 \prime +3m_2 \prime$$. Maybe I'm forgetting something from grade school algebra, but I have no idea how to solve for $$m_1 \prime, m_2 \prime$$. I've plugged it into wolfram alpha [http://www.wolframalpha.com/input/?i=2*x_1+3*x_2+++1+=+2*y_1+++3*y_2] and got solutions (all I care about is the case n =1 for the integer solutions wolfram gives) but I want to know how to arrive there.

2. Sep 22, 2011

### CompuChip

If you have to add one to a number, how could you do that if you only have increments / decrements of 2 and 3 available?