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Inductive Proof (+linear equation in four variables)

  1. Sep 21, 2011 #1
    I'm trying to prove by induction that [tex]\forall n \geq 5, \exists m_1, m_2 \in \mathbb{N} [/tex] such that [tex] n = 2m_1 +3m_2. [/tex]I've done the base case, and the inductive step boils down to showing that [tex] \exists m_1 \prime m_2 \prime [/tex] such that [tex]2m_1 +3m_2 +1 = 2m_1 \prime +3m_2 \prime [/tex]. Maybe I'm forgetting something from grade school algebra, but I have no idea how to solve for [tex] m_1 \prime, m_2 \prime [/tex]. 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. jcsd
  3. Sep 22, 2011 #2


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    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?
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