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 Harmonic Diophantine Hindrance (Posted on 2015-06-21)
A, B and C are three strictly increasing positive integers in this order such that B is the harmonic mean of A and C.
Can A2 + B2 = C2 + 1?
If so, provide an example.
If not, prove it.

 No Solution Yet Submitted by K Sengupta No Rating

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 re: question re: Possible solution Comment 3 of 3 |
(In reply to question re: Possible solution by xdog)

Answer: I was called away half-way through before I could finish.

c = (ab/(2a-b))

Now we have a^2+b^2= (ab/(2a-b))^2+1

a cannot be 1, for then b=c.

Say a is much larger than b:
Then  (a/(2a))^2 is around (a^2)-1 involving the contradiction that a is less than 2.
Say b is much larger than a:
Then (b/(-b))^2 is around (b^2)-1  involving the contradiction that b is less than 2.
Say a and b are around the same size
(a^2/(a))^2 is around (2a^2-1)
and a is still less than 2.

So a can't be 1, yet can't be large enough to be greater than 1.

 Posted by broll on 2015-06-23 00:45:51

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