a step beyond Euclid and Fermat, 5

For the sake of reference, you may like to open the previous note, step 4, in a separate window.

Let

Par(n)  :=  { k : 0 ≤ k ≤ n and k = n mod 2}

be the set of all non-negative numbers k ≤ n, which have the same parity as n; e.g.

• Par(0) = {0}
• Par(1) = {1}
• Par(2} = {0 2}
• Par(3) = {1 3}
• Par(4) = {0 2 4}

etc.

It follows from the initial values of the sequences (g(n) and (A(n), and from the last equality of "step 4" that:

• A(n)  =  ∏_{k ∈ Par(n)} g(k)     ∀ n=1 2...
  

The first three terms, g(0) g(1) g(2), of sequence (g(n)), are pairwise relatively prime.

If every two of the terms g(0) ... g(n) (with different indices) are relatively prime then obviously A(n-1) and A(n) are relatively prime too. Since, by the very definition, g(n+1) is the difference of A(n) and A(n-1), g(n+1) is relatively prime with A(n-1) and with A(n), which means that g(n+1) is relatively prime to every previous term g(0) ... g(n). This holds for every n=2 3 ... Thus every two different terms of sequence (g(n)) are relatively prime.