mathematics

Taxicab number

Taxicab number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Taxicab number rather than just read about it. In short: In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy–Ramanujan number.

Key takeaways

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

Reference excerpt

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy–Ramanujan number. The name is derived from a conversation ca. 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

History and definition The pairs of summands of the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned by Bernard Frénicle de Bessy, who published his observation in 1657. 1729 was made famous as the first taxicab number in the early 20th century by a story involving Srinivasa Ramanujan in claiming it to be the smallest for his particular example of two summands. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims about whether the thus-generated numbers are the smallest possible and so it does not directly provide the actual value of Ta(n). The taxicab numbers subsequent to 1729 were found with the help of computers. John Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1989. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999. Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008, following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6). Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006. The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. (sequence A293647 in the OEIS). The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.

Known taxicab numbers So far, the following 6 taxicab numbers are known:

… excerpt ends here. Continue reading the full article.

Illustrations

Taxicab number: Srinivasa Ramanujan (picture) was bedridden when he developed the idea of taxicab numbers, according to an anecdote from G. H. Hardy. — Srinivasa Ramanujan (picture) was bedridden when he developed the idea of taxicab numbers, according to an anecdote from G. H. Hardy.

Worked examples

Example 1 — a first encounter with Taxicab number

Start with the simplest possible case. Write down what Taxicab number 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 Taxicab number 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 Taxicab number 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 Taxicab number

In research: Taxicab number 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 Taxicab number 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: Taxicab number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Number theory, Srinivasa Ramanujan, so understanding it makes those chapters shorter.
In everyday life: Look for Taxicab number 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.

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How to study Taxicab number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Taxicab number 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 Taxicab number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Taxicab number in simple terms?

Why does Taxicab number 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 Taxicab number?

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 Taxicab number.