If a 2 + b b 2 + c is rational, then a + b...

amacorrit80

amacorrit80

Answered

2022-07-16

If a 2 + b b 2 + c is rational, then a + b + c | a b + b c + c a
Let a , b , c N > 0 . Prove that if
a 2 + b b 2 + c
is a rational number, then a + b + c | a b + b c + c a
Hints?
My approach was to assume that
a 2 + b b 2 + c = p q
then
p b 2 + p c = q a 2 + q b
or
2 ( p b q a ) = q b p c
squaring:
2 ( p b q a ) 2 = ( q b p c ) 2
but that doesn't seems to do much.

Answer & Explanation

Kali Galloway

Kali Galloway

Expert

2022-07-17Added 16 answers

Let a 2 + b b 2 + c = r Q
Hence, ( a r b ) 2 = r c b
If a r b 0 so we get a contradiction.
Thus, a r b = 0 and r c b = 0 or a = r b and c = b r
Id est,
a b + a c + b c a + b + c = b 2 ( 1 + r + 1 r ) b ( 1 + r + 1 r ) = b
and we are done!

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