gordonm888
Joined: Feb 18, 2015
• Posts: 2823
September 11th, 2019 at 8:12:27 PM permalink
Quote: ThatDonGuy

I started a brute force search, and could not find any n < 16,000 such that n2 + 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"

Edit: I found it.
Last edited by: gordonm888 on Sep 11, 2019
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
Joined: Feb 18, 2015
• Posts: 2823
Thanks for this post from:
September 12th, 2019 at 11:59:41 AM permalink
I'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

I can also see why n2+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!
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4639
September 12th, 2019 at 5:08:10 PM permalink
Quote: gordonm888

I'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 n2 + 2.
kubikulann
Joined: Jun 28, 2011