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Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some...

Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some positive integer n, such that every pair an, am (for n != m) are relatively prime. [Hint: Assume there exists a finite sequence an1 < an2 < an3 < . . . < anm, where nj are increasing positive integers. Show that using those numbers we can construct a new number that fulfills the requirements.]

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Answer #1

We have to show there is two natural numbers

n and m , such that an and am these two sequences of numbers are relatively prime that is gcd ( an , am ) = 1

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