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If
where
Then
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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