mathematics

Nonhypotenuse number

Nonhypotenuse 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 Nonhypotenuse number rather than just read about it. In short: In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides.

Key takeaways

Nonhypotenuse 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 Nonhypotenuse number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nonhypotenuse number from memory before moving on to harder problems.

Reference excerpt

In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides. The numbers 1, 2, 3, and 4 are all nonhypotenuse numbers. The number 5, however, is not a nonhypotenuse number as 5 2 = 3 2 + 4 2 {\displaystyle 5^{2}=3^{2}+4^{2}} . The first fifty nonhypotenuse numbers are:

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84 (sequence A004144 in the OEIS) Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value x scales asymptotically with x/√log x. The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1. Equivalently, they are the number that cannot be expressed in the form K ( m 2 + n 2 ) {\displaystyle K(m^{2}+n^{2})} where K, m, and n are all positive integers. A number whose prime factors are not all of the form 4k+1 cannot be the hypotenuse of a primitive integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle. The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first n {\displaystyle n} square numbers using only n + o ( n ) {\displaystyle n+o(n)} additions.

See also Pythagorean theorem Landau-Ramanujan constant Fermat's theorem on sums of two squares

References

External links OEIS sequence A004144 (Nonhypotenuse numbers) OEIS sequence A125667 (Eta numbers)

Illustrations

Worked examples

Example 1 — a first encounter with Nonhypotenuse number

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

In research: Nonhypotenuse 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 Nonhypotenuse 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: Nonhypotenuse number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Pythagorean theorem, so understanding it makes those chapters shorter.
In everyday life: Look for Nonhypotenuse 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Nonhypotenuse number” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Nonhypotenuse number in 20 minutes

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

Frequently asked questions

What is Nonhypotenuse number in simple terms?

Why does Nonhypotenuse 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 Nonhypotenuse 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 Nonhypotenuse number.