[seqfan] Fun little identity (to me anyway)

David Wilson dwilson at gambitcomm.com
Wed Jan 21 15:32:35 CET 2009

While lying in bed last night, I kept myself occupied working on a sequence

     a(1) = 1;
    a(n+1) = smallest value > a(n) with a(n)a(n+1) of form k^2-2.

I got as far as


when I realized that after 1, these numbers were themselves of the form
k^2-2.  I was able to formulate

    (n^2-2)((n+1)^2-2) = ((n(n+1)-2)^2-2)

which neatly shows that the product of two adjacent numbers of the form
k^2-2 is also of that form.

So I looked at k^2-3, and to my surprise

    (n^2-3)((n+1)^2-3) = ((n(n+1)-3)^2-3)

held as well, and I immediately generalized to

    (n^2+k)((n+1)^2+k) = ((n(n+1)+k)^2+k)

This was a very pleasant result. The product of two adjacent numbers of
the form n^2+k is again of that form. The subexpression n(n+1)+k is a
added bonus.

I'm sure I'm not the first discoverer, I imagine this is a minor footnote in
quadratic field theory or something, but it kept me amused in the darkness.

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