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gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 32
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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"



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

Edit: I found it.
Last edited by: gordonm888 on Sep 11, 2019
Sometimes, people are just a bottomless mystery. And, after all, this is just a sh*tty little forum in the sun-less backwaters of the online world.
gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 32
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Thanks for this post from:
kubikulann
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!
Sometimes, people are just a bottomless mystery. And, after all, this is just a sh*tty little forum in the sun-less backwaters of the online world.
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 87
  • Posts: 3719
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
kubikulann
Joined: Jun 28, 2011
  • Threads: 27
  • Posts: 890
September 12th, 2019 at 5:34:40 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!

Delightful and fascinating. I love learning that stuff. Thank you both.
Reperiet qui quaesiverit

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