- #1
anemone
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The sum of three integers $x,\,y,\,z$ is zero. Show that $2x^4+2y^4+2z^4$ is the square of an integer.
Prove 2x⁴+2y⁴+2z⁴ is the square of an integer
In summary, the sum of three integers $x,\,y,\,z$ is zero and it is shown that $2x^4+2y^4+2z^4$ is the square of an integer.
- #2
MarkFL
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My solution:
We have:
\(\displaystyle z=-(x+y)\)
Hence:
\(\displaystyle S=2x^4+2y^4+2z^4=2\left(x^4+y^4+(x+y)^4\right)\)
\(\displaystyle S=2\left(x^4+y^4+x^4+4x^3y+6x^2y^2+4xy^3+y^4\right)\)
\(\displaystyle S=2\left(2x^4+2y^4+4x^3y+6x^2y^2+4xy^3\right)\)
\(\displaystyle S=4\left(x^4+2x^3y+3x^2y^2+2xy^3+y^4\right)\)
\(\displaystyle S=4\left(x^2+xy+y^2\right)^2\)
\(\displaystyle S=\left(2\left(x^2+xy+y^2\right)\right)^2\)
If $x$ and $y$ are integers, then S must be the square of an integer.
\(\displaystyle z=-(x+y)\)
Hence:
\(\displaystyle S=2x^4+2y^4+2z^4=2\left(x^4+y^4+(x+y)^4\right)\)
\(\displaystyle S=2\left(x^4+y^4+x^4+4x^3y+6x^2y^2+4xy^3+y^4\right)\)
\(\displaystyle S=2\left(2x^4+2y^4+4x^3y+6x^2y^2+4xy^3\right)\)
\(\displaystyle S=4\left(x^4+2x^3y+3x^2y^2+2xy^3+y^4\right)\)
\(\displaystyle S=4\left(x^2+xy+y^2\right)^2\)
\(\displaystyle S=\left(2\left(x^2+xy+y^2\right)\right)^2\)
If $x$ and $y$ are integers, then S must be the square of an integer.
- #3
anemone
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(Yes) Well done, MarkFL! So, do you want a cup of coffee or me singing a lullaby for you? Hehehe...:pMarkFL said:
- #4
kaliprasad
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I feel I also deserve a cup of coffee as well because
We have
$(x^2 + y^2 + z^2)^2 = x^4 + y^4 + z^4 + 2x^2y^2 + 2 y^2 z^2 + 2 z^2 x^2 \cdots 1$
Now $(x^2 y^2 + x^2 z^2) = x^2((y+z)^2 – 2yz) = x^4 – 2x^2yz \dots2$ (as y+z = - x)
Similarly
$y^2 z^2 + y^2 x^2 = y^4 – 2y^2xz \cdots 3$
$z^2x^2 +z^2y^2 = z^4 – 2z^2xy \cdots4$
from (2) (3) and (4)
$2(x^2y^2 + y^2 z^2 + z^2 x^2) = (x^4 + y^4 + z^4 – 2xyz(x+y+z))$
= $x^4 + y^4 + z^4 ...5$
as x+y+z = 0
Putting value of $2(x^2y^2 + y^2 z^2 + z^2 x^2)$ from (5) in (1) we get the result
$2(x^4+y^4+z^4)= (x^2+y^2+z^2)^2$
$(x^2 + y^2 + z^2)^2 = x^4 + y^4 + z^4 + 2x^2y^2 + 2 y^2 z^2 + 2 z^2 x^2 \cdots 1$
Now $(x^2 y^2 + x^2 z^2) = x^2((y+z)^2 – 2yz) = x^4 – 2x^2yz \dots2$ (as y+z = - x)
Similarly
$y^2 z^2 + y^2 x^2 = y^4 – 2y^2xz \cdots 3$
$z^2x^2 +z^2y^2 = z^4 – 2z^2xy \cdots4$
from (2) (3) and (4)
$2(x^2y^2 + y^2 z^2 + z^2 x^2) = (x^4 + y^4 + z^4 – 2xyz(x+y+z))$
= $x^4 + y^4 + z^4 ...5$
as x+y+z = 0
Putting value of $2(x^2y^2 + y^2 z^2 + z^2 x^2)$ from (5) in (1) we get the result
$2(x^4+y^4+z^4)= (x^2+y^2+z^2)^2$
- #5
MarkFL
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anemone said:
Hmmm...one is a stimulant and the other a sedative...so perhaps I should have both so they will counteract one another. :D
- #6
anemone
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kaliprasad said:
Of course you do! This is what I prepared for you, kali, my friend!
And this is for my sweetest admin, MarkFL!:D
Related to Prove 2x⁴+2y⁴+2z⁴ is the square of an integer
1. How do you prove that 2x⁴+2y⁴+2z⁴ is the square of an integer?
To prove that 2x⁴+2y⁴+2z⁴ is the square of an integer, we need to show that it can be expressed as the product of two integers. This can be done through algebraic manipulation and factoring.
2. Can you provide an example of how to prove 2x⁴+2y⁴+2z⁴ is the square of an integer?
An example of proving 2x⁴+2y⁴+2z⁴ is the square of an integer would be to take the expression and factor out a common factor of 2, leaving us with 2(x⁴+y⁴+z⁴). From there, we can factor the remaining polynomial as the difference of two squares, giving us 2(x²+y²+z²)(x²-y²+z²). This can then be simplified to the product of two integers, making it the square of an integer.
3. Are there any specific conditions that need to be met for 2x⁴+2y⁴+2z⁴ to be the square of an integer?
In order for 2x⁴+2y⁴+2z⁴ to be the square of an integer, x, y, and z must all be integers themselves. If any of these variables are not integers, then the expression cannot be the square of an integer.
4. What are some common techniques used for proving that an expression is the square of an integer?
Some common techniques used for proving that an expression is the square of an integer include algebraic manipulation, factoring, and using properties of square numbers. In some cases, using specific examples and showing that they satisfy the given expression can also be a valid proof.
5. Why is proving that 2x⁴+2y⁴+2z⁴ is the square of an integer important in mathematics?
Proving that 2x⁴+2y⁴+2z⁴ is the square of an integer is important in mathematics because it allows us to understand and prove important concepts related to square numbers and polynomials. It also helps us develop problem-solving skills and critical thinking in algebraic contexts.
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