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

lurtzslikgtgjd 2022-05-08 Answered
Let A be the set of all d such that d < 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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Answers (1)

Madelyn Lynch
Answered 2022-05-09 Author has 15 answers
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 > 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 ', 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 < p 2 1 for p > 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') and so it's different for every p. Since there are infinitely many 's p, the set A is infinitely large (it actually contains a lot more numbers than we've shown, but that is irrelevant for it infinite-ness).
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