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#1 2011-11-22 18:08:56

jacks
Guest

gcd

If

where

Then

#2 2011-11-22 19:41:22

reconsideryouranswer
Member
Registered: 2011-05-11
Posts: 171

Re: gcd

jacks wrote:

If

where

Then

As a consequence, where an n value works, it must be even,
because n^2 + 1 and n^2 + n will be even when n is odd
and hence divisible by at least 2.

Whenever n^2 + 1 is a prime (greater than 2), then that n is a solution.

It is not known whether there are an infinite number of primes
of the form n^2 + 1.

Source:
http://mathworld.wolfram.com/PrimeNumber.html


If n^2 + 1 equals certain composite composite numbers, then it
is possible for the gcd = 1 for those certain cases.

An example is n = 8.  Then n^2 + 1 = 65 (composite)
and n^2 + n = 72, which is also composite, but the
gcd(65, 72) = 1 and therefore n = 8 works.


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