Quote:ThatDonGuyI started a brute force search, and could not find any n < 16,000 such that n

^{2}+ 2 was a multiple of any number of the form 8k + 7.

EDIT: None found through 25,000

I have also asked the math boffins at Art of Problem Solving to see if they can come up with anything.

EDIT: Somebody has come up with a solution (that confirms the conjecture) that I need to check; it involves "quadratic residues" and "Legendre symbols"

Can you post a link? I can read Legendre symbols and quadratic residues.

Edit: I found it.

Proof: Read Post #14 and subsequent Posts by Stormersyle

I can also see why n

^{2}+3 would be divisible by all of the congruence classes of 8, i.e., by 1,3,5,7(mod 8).

I have submitted a comment on this to the OEIS website and it appears to have been accepted for 'publication.'

Thanks to kubikulann and ThatDonGuy for your help with this. Two of the smartest guys on this forum!

Quote:gordonm888I'm satisfied with the proof that ThatDonGuy stimulated on The Art of Problem Solving site. You can read it here:

Proof: Read Post #14 and subsequent Posts by Stormersyle

Also note that the same proof shows that no integer congruent with 5 (mod 8) is a factor of n

^{2}+ 2.

Delightful and fascinating. I love learning that stuff. Thank you both.Quote:gordonm888I'm satisfied with the proof that ThatDonGuy stimulated on The Art of Problem Solving site. You can read it here:

Proof: Read Post #14 and subsequent Posts by Stormersyle!