Sums of three cubes

Key takeaways

Sums of three cubes belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sums of three cubes to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sums of three cubes from memory before moving on to harder problems.

Reference excerpt

In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer n {\displaystyle n} to equal such a sum is that n {\displaystyle n} cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient. Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non-zero natural density.

Small cases A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's Last Theorem for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefore, by Leonhard Euler's proof of that case of Fermat's last theorem, there are only the trivial solutions

a 3 + ( − a ) 3 + 0 3 = 0. {\displaystyle a^{3}+(-a)^{3}+0^{3}=0.}

For representations of 1 and 2, there are infinite families of solutions

( 9 b 4 ) 3 + ( 3 b − 9 b 4 ) 3 + ( 1 − 9 b 3 ) 3 = 1 {\displaystyle (9b^{4})^{3}+(3b-9b^{4})^{3}+(1-9b^{3})^{3}=1} (discovered by K. Mahler in 1936) and

( 1 + 6 c 3 ) 3 + ( 1 − 6 c 3 ) 3 + ( − 6 c 2 ) 3 = 2 {\displaystyle (1+6c^{3})^{3}+(1-6c^{3})^{3}+(-6c^{2})^{3}=2} (discovered by A.S. Verebrusov in 1908, quoted by L.J. Mordell). These can be scaled to obtain representations for any cube or any number that is twice a cube. There are also other known representations of 2 that are not given by these infinite families:

1 214 928 3 + 3 480 205 3 + ( − 3 528 875 ) 3 = 2 , {\displaystyle 1\ 214\ 928^{3}+3\ 480\ 205^{3}+(-3\ 528\ 875)^{3}=2,}

37 404 275 617 3 + ( − 25 282 289 375 ) 3 + ( − 33 071 554 596 ) 3 = 2 , {\displaystyle 37\ 404\ 275\ 617^{3}+(-25\ 282\ 289\ 375)^{3}+(-33\ 071\ 554\ 596)^{3}=2,}

3 737 830 626 090 3 + 1 490 220 318 001 3 + ( − 3 815 176 160 999 ) 3 = 2. {\displaystyle 3\ 737\ 830\ 626\ 090^{3}+1\ 490\ 220\ 318\ 001^{3}+(-3\ 815\ 176\ 160\ 999)^{3}=2.}

However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials as above. Even in the case of representations of 3, Louis J. Mordell wrote in 1953 "I do not know anything" more than its small solutions

1 3 + 1 3 + 1 3 = 4 3 + 4 3 + ( − 5 ) 3 = 3 {\displaystyle 1^{3}+1^{3}+1^{3}=4^{3}+4^{3}+(-5)^{3}=3}

and the fact that each of the three cubed numbers must be equal modulo 9.

Computational results Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations. Elsenhans & Jahnel (2009) used a method of Noam Elkies (2000) involving lattice reduction to search for all solutions to the Diophantine equation

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Sums of three cubes

Start with the simplest possible case. Write down what Sums of three cubes claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sums of three cubes before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sums of three cubes ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sums of three cubes

In research: Sums of three cubes appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sums of three cubes in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sums of three cubes is common in secondary-school and first-year university syllabi. It links to neighbouring topics Additive number theory, Diophantine equations, Louis J. Mordell, so understanding it makes those chapters shorter.
In everyday life: Look for Sums of three cubes outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Sums of three cubes in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sums of three cubes means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sums of three cubes out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sums of three cubes in simple terms?

In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer n {\displaystyle n} to equal such a sum is that n {\displayst…

Why does Sums of three cubes matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sums of three cubes?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sums of three cubes.