Hi everyone! Hopefully you are fine today. Not long until 2013. And yes, I am 99.44% confident that the human race will still be on Earth be here come 12/22/2012.
x + x^2 = n^2
Let n,x ∈ N. N represents the natural numbers. Natural numbers are commonly referred to the whole numbers 1, 2, 3, etc. Some mathematicians include 0.
Are there any integer solutions to x + x^2 = n^2 with n,x ∈ N?
Our first instinct is most likely to go grab the nearest calculator or computer. Observe that:
(I). x + x^2 = n^2
(II). x^2 * (1/x + 1) = n^2
Taking the (principal) square root of both sides yields:
(III). x * √(1/x + 1) = n
In order for (III) to be true:
1. The quantity 1/x + 1 has to be an integer,
2. 1/x + 1 has to be a perfect square, and
3. x = 1/x + 1
The only natural number that allows condition 1 to be true is when x = 1.
Then 1/x + 1 = 2.
We know that 2 is not a perfect square (√2 ≈ 1.41421), so condition 2 fails. (II) does not fit because 1 * √2 = √2, leaving n = √2, not fitting the requirement that n,x ∈ N.
According to this analysis, there are no natural number solutions to x + x^2 = n^2.
x + x^2 + x^3 = n^q
Let n,x ∈ N. Suppose q = 2. Can we find solutions with these conditions?
x + x^2 + x^3 = n^2
x^2 * (1/x + 1 + x) = n^2
x * √(1/x + 1 + x) = n
Once again the only way 1/x + 1 + x is an integer is that when x = 1. However when x=1, 1/x + 1 + x = 3, and √3 ≈ 1.73205. And 1 * √3 = √3, n = √3, which is not a natural number.
There are no solutions (in the natural number set) for x + x^2 + x^3 = n^2.
What about q = 3?
Then x + x^2 + x^3 = n^3
x^3 * (1 + 1/x + 1/(x^2)) = n^3
x * ∛(1 + 1/x + 1/(x^2)) = n
Again, the only possible candidate is when x=1, but that leaves n = ∛3. No solutions in the natural number set.
Not all is "lost"...
Consider trying to find solutions to:
x + x^2 + x^3 + x^4 = n^2 where n,x ∈ N.
Then x * √(1/x + 1 + x + x^2) = n.
If x = 1, then √(1/x + 1 + x + x^2) = √4 = 2, and n = 1 * 2 = 2. Success! I believe x=1 and n=2 is the only solution to this equation with these conditions imposed.
Happy Holidays everyone and see you next time!
Eddie
This blog is property of Edward Shore. 2012
A blog is that is all about mathematics and calculators, two of my passions in life.
Wednesday, December 19, 2012
Equations without Whole Number Solutions  Fruitless Search?
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