Multiply -2x²y(y²x² - x²z + xy³) and verify the result when x=2, y= -1, z=-2
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6 November 2019 · Facebook for Android ·
Humankind have been studying numbers for thousands of years, you might think we know everything about the number 3.
But mathematicians recently discovered something new about 3: a third way to express it as the sum of three cubes. Expressing a number as the sum of three perfect cubes is a surprisingly interesting problem. It’s easy to show that most numbers can’t be written as one cube or the sum of two cubes, but it’s conjectured that most numbers can in fact be written as the sum of three cubes. Finding those three cubes, however, can be quite a challenge.
For example, we knew we could write 3 as 1³ + 1³ + 1³ and also as 4³ + 4³ + (−5)³, but for over 60 years mathematicians wondered if there was another way. This past September, Andrew Booker and Andrew Sutherland finally found a third solution:
3 = 569,936,821,221,962,380,720³ + (−569,936,821,113,563,493,509)³ + (−472,715,493,453,327,032)³
What about writing 33 as a sum of three squares?
We would need to find three “perfect squares” — numbers that are equal to an integer times itself.
Suppose we want to search for a solution to:
33 = x³ + y³ + z³
Above equation can be written as:
33 – (x³ + y³) = z³
Instead of running through all the triples (x, y, z), we will run through the pairs (x, y).
For each pair, we compute and then check a list of perfect cubes to see if our result (z³) is on it. If it is, we’ve found a combination that works. If it isn’t, we keep looking. This substantially reduces the size of our search space: Instead of the 8,000,00 triples (x, y, z), we’re now searching the 200 x 200 = 40,000 pairs (x, y). It’s a big savings, but it’s still not enough to make finding a solution computationally feasible.
An even better approach is to rewrite the equation like this:
33 – z³ = x³ + y³
Now we search through the z’s. For each z, we compute, and then we use a neat little trick from math class. The expression can always be factored in the following way:
x³ + y³ = (x + y)(x² – xy + y²)
This is known as the sum-of-cubes formula.
To verify this, we just multiply out the right side using the distributive property:
(x + y)(x² – xy + y²)
= x³ – x²y + xy² + yx² – xy² + y³
= x³ + y³
How does this formula help us in our search?
Once we’ve computed 33 – z³, we factor it into primes, which is something computers are pretty good at, at least in the range of numbers we’re looking at. And once we’ve factored 33 – z³, we check if the factors can be arranged like (x + y)(x² – xy + y²). If they can, we’ve found a solution.
For example, let’s say we were trying to find a way to write the number 34 as a sum of three cubes, and our search led us to z = −6. We compute 34 – z³ = 34 – (-6)³ = 34 – (-216) = 34 + 216 = 250, and then we see how we can factor 250.
After some investigating, we realize that we can write 250 = 10 × 25 = (5+5)(5² – 5 × 5 + 5²). This is exactly (x + y)(x² – xy + y²) for x = 5 and y = 5, so the triple (x, y, z) = (5, 5, -6) should work for 34.
Sure enough, 34 = 5³ + 5³ + (-6)³, and we’ve successfully found three cubes that sum to 34.