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Prime and Twice Square (Posted on 2016-10-22) Difficulty: 4 of 5
Determine all possible values of a prime number P such that each of P+1 and P2 + 1 is twice a perfect square.

Prove that these are the only ones.

No Solution Yet Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

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Some Thoughts re: A solution | Comment 3 of 4 |
(In reply to A solution by Math Man)

No prime other than the one already mentioned has been found under 18.8 million (that's where my program was stopped). But of course that's not a proof.


 Do
   p = nxtprm(p)
   DoEvents
   pp1 = p + 1
   sq = pp1 / 2
   sr = Int(Sqr(sq) + 0.5)
   If sr * sr = sq Then
      sr1 = sr
      p2p1 = p * p + 1
      sq = p2p1 / 2
      sr = Int(Sqr(sq) + 0.5)
      If sr * sr = sq Then
         Text1.Text = Text1.Text & p & Str(p + 1) & Str(p * p + 1) & "         "
         Text1.Text = Text1.Text & Str((p + 1) / 2) & Str((p * p + 1) / 2) & "      "
         Text1.Text = Text1.Text & sr1 & Str(sr) & crlf
      End If
   End If
 Loop
 


  Posted by Charlie on 2016-10-22 15:54:41
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