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Figuring Forty-Nine and Sequence (Posted on 2016-10-25) Difficulty: 3 of 5
Find all possible triples (P, Q, R) of positive integers satisfying both of the following conditions

49, P2 and Q2 (in order) are in arithmetic sequence with common difference > 0 , and:

49, Q2 and R2 (in order) are in arithmetic sequence with common difference > 0

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts No proof, but ... | Comment 1 of 2
                                   differences
 p  q  r     p^2 q^2 r^2      1st seq     2nd seq
13 17 23     169 289 529      120 120     240 240

any others would have p > 10,000,000

 For p = 8 To 10000000
   p2 = p * p
   q2 = p2 + p2 - 49
   sr = Int(Sqr(q2) + 0.5)
   q = sr
   If sr * sr = q2 Then
      r2 = q2 + q2 - 49
      sr = Int(Sqr(r2) + 0.5)
      r = sr
      If sr * sr = r2 Then
        Text1.Text = Text1.Text & p & Str(q) & Str(r) & "     "
        Text1.Text = Text1.Text & p2 & Str(q2) & Str(r2) & "     "
        Text1.Text = Text1.Text & Str(p2 - 49) & Str(q2 - p2) & "    " & Str(q2 - 49) & Str(r2 - q2) & crlf
      End If
   End If
 Next


  Posted by Charlie on 2016-10-25 09:18:28
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