perplexus.info :: Sequences : Figuring Forty-Nine and Sequence

Figuring Forty-Nine and Sequence (Posted on 2016-10-25)

Find all possible triples (P, Q, R) of positive integers satisfying both of the following conditions

49, P²and 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.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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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

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

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