mathematics

Odious number

Odious 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 Odious number rather than just read about it. In short: In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers.

Key takeaways

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

Reference excerpt

In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers. In computer science, an odious number is said to have odd parity.

Examples The first odious numbers are 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, and so on.

Properties If a ( n ) {\displaystyle a(n)} denotes the n {\displaystyle n} th odious number (with a ( 0 ) = 1 {\displaystyle a(0)=1} ), then for all n {\displaystyle n} , a ( a ( n ) ) = 2 a ( n ) {\displaystyle a(a(n))=2a(n)} . Every positive integer n {\displaystyle n} has an odious multiple that is at most n ( n + 4 ) {\displaystyle n(n+4)} . The numbers for which this bound is tight are exactly the Mersenne numbers with even exponents, the numbers of the form n = 2 2 r − 1 {\displaystyle n=2^{2r}-1} , such as 3, 15, 63, etc. For these numbers, the smallest odious multiple is exactly n ( n + 4 ) = 2 4 r + 2 2 r + 1 − 3 {\displaystyle n(n+4)=2^{4r}+2^{2r+1}-3} .

Related sequences The odious numbers give the positions of the non-zero values in the Thue–Morse sequence. Every power of two is odious, because its binary expansion has only one non-zero bit. Except for 3, every Mersenne prime is odious, because its binary expansion consists of an odd prime number of consecutive non-zero bits. Non-negative integers that are not odious are called evil numbers. The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pair-wise sums.

References

External links

Weisstein, Eric W., "Odious Number", MathWorld

Illustrations

Worked examples

Example 1 — a first encounter with Odious number

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

In research: Odious 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 Odious 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: Odious number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Odious 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 Odious number in 20 minutes

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

Frequently asked questions

What is Odious number in simple terms?

In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers.

Why does Odious 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 Odious 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 Odious number.