hello, I need help with the following problem. I know that a,b,c,x are integers and that 2(a^2 + b^2 + c^2) = 5(ab+bc+ca)=x, I have to prove that 10x is a square of some integer. Can someone help?
#algebra or numbertheory question
potent thorn
grand heron
ok ill be honest
idrk where to start
but I would start by multiplying the whole thing by 10
to get 20(a^2 + b^2 + c^2) = 50(ab + bc + ca) = 10x
for the middle section
if you try to expand and refactorise
you can get 50ab + 50bc + 50ca
im sorry idk
hope this can start u somewhere
thorny aurora
I don't sure, but have you tried vieta jumping?
thorny aurora
Is it true for many tuples?
potent thorn
then I would have to somehow prove that 10/3(a+b+c) is an integer
potent thorn
Is it true for many tuples?
what do you mean? it is true because of the conditions given in the question
thorny aurora
nvm, I also claimed that, but I haven't proven it
potent thorn
9x = 20(ab + bc + ca) + 10(a^2 + b^2 + c^2)=10(a+b+c)^2
grand shuttleBOT
k12byda5h
thorny aurora
therefore 3|a+b+c
potent thorn
oh wait lol
potent thorn
therefore 3|a+b+c
how did you get that
thorny aurora
**k12byda5h**
9|2(a+b+c)^2
but gcd(2,9) = 1
so 9|(a+b+c)^2
and 3|9|(a+b+c)^2
since 3 is a prime
it also implies that 3|a+b+c
grand heron
$2(a+b+c)^2 = 9(ab+bc+ca)$
how did u get this mate?
thorny aurora
add 4ab+4bc+4ac to both sides of the equation
grand heron
ok
thorny aurora
both sides
I'm faster 
potent thorn
lol
grand heron
how does that help tho
potent thorn
2(a^2 + b^2 + c^2) + 4(ab + bc +ca) = 2(a+b+c)^2
thorny aurora
but I got that 10x = (10/3(a+b+c))^2
we already have that the expressions are equal to $10\left(\frac{a+b+c}3\right)^2$
grand shuttleBOT
k12byda5h
potent thorn
9x = 20(ab + bc + ca) + 10(a^2 + b^2 + c^2)=10(a+b+c)^2
here
grand heron
tf is d?
potent thorn
I meant x
thorny aurora
how did u get from 5(ab+bc+ca)
I guess you can compute by yourself, it is just bashing
potent thorn
9x = 4x + 5x, 4x = 20(ab + bc + ca) and 5x = 10(a^2 + b^2 + c^2)
it is basically given in the problem
so 9x = 10(a+b+c)^2
which implies that x =
potent thorn
**k12byda5h**
here