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Old 2019-01-05, 00:25   #2
science_man_88
 
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Jul 2009
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Quote:
Originally Posted by enzocreti View Post
Consider the equation

a*(2*a^2+2*b^2+c^2+1)=(2*a^3+2*b^3+c^3+1)

with a,b,c positive integers.
Are the only solutions to that equation a=1, b=1, c=1 and a=5, b=4 and c=6?
multiplying each side through gives:

\[2a^3+2ab^2+ac^2+a=2a^3+2b^3+c^3+1\]

which then cancels down to:

\[2ab^2+ac^2+a=2b^3+c^3+1\]

which with a=b=c=1 goes to:

\[2b^3+c^3+1=2b^3+c^3+1\]

So no, there are multiple solutions. That being said, you should be able to work this out on your own, before you get taken seriously. okay sorry didn't see you listed a=c=b=1 . You could try algebraic relations between variables. then you can use them to go to univariate polynomials and apply polynomial remainder theorem.

Last fiddled with by science_man_88 on 2019-01-05 at 00:45
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