Let A be the set of all d such that d &lt; 1 and d = a b where a 2 + b 2 = c 2 and a , b , c ∈ I. Is A an infinite or finite set and how can that be proven?

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Let     A be the set of all   d such that   d  &amp;lt;  1 and   d  =      a    b   where       a    2    +      b    2    =      c    2   and   a  ,  b  ,  c  ∈  I. Is A an infinite or finite set and how can that be proven?

Madelyn Lynch · Accepted Answer

Well, Pythagorean triples are given by  (  a  ,  b  ,  c  )  =  (  2  p  q  ,      p    2    −      q    2    ,      p    2    +      q    2    )for integers   p and   q with   p  &amp;gt;  q (to prove that these are all is irrelevant; we only need infinitely many triples, not necessarily all of them).So let   q  =  1 and   p an odd &#039;, then we have the triple   (  2  p  ,      p    2    −  1  ,      p    2    +  1  ). The corresponding   d would then be   d  =            2      p                      p        2            −      1       (since   2  p  &amp;lt;      p    2    −  1 for   p  &amp;gt;  2), this is an irreducible fraction if we divide both sides by   2 (that is       p          (              p        2            −      1      )          /      2      , and this is irreducible because       p    2    −  1 and   p are co&#039;) and so it&#039;s different for every   p. Since there are infinitely many &#039;s   p, the set   A is infinitely large (it actually contains a lot more numbers than we&#039;ve shown, but that is irrelevant for it infinite-ness).

Let A be the set of all d such that d < 1 and d = a b </mfr

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