mathematics

Highly cototient number

Highly cototient 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 Highly cototient number rather than just read about it. In short: In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k} than any other integer below k {\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totient function.

Key takeaways

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

Reference excerpt

In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation

x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k}

than any other integer below k {\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totient function. There are infinitely many solutions to the equation for

k {\displaystyle k} = 1 so this value is excluded in the definition. The first few highly cototient numbers are:

2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... (sequence A100827 in the OEIS) Many of the highly cototient numbers are odd. The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factorization becomes harder as the numbers get larger.

Example The cototient of x {\displaystyle x} is defined as x − ϕ ( x ) {\displaystyle x-\phi (x)} , i.e. the number of positive integers less than or equal to x {\displaystyle x} that have at least one prime factor in common with x {\displaystyle x} . For example, the cototient of 6 is 4 since these four positive integers have a prime factor in common with 6: 2, 3, 4, 6. The cototient of 8 is also 4, this time with these integers: 2, 4, 6, 8. There are exactly two numbers, 6 and 8, which have cototient 4. There are fewer numbers which have cototient 2 and cototient 3 (one number in each case), so 4 is a highly cototient number. (sequence A063740 in the OEIS)

Primes The first few highly cototient numbers which are primes are

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889, 2099, 2309, 2729, 3359, 3989, 4289, 4409, 5879, 6089, 6719, 9029, 9239, ... (sequence A105440 in the OEIS)

Worked examples

Example 1 — a first encounter with Highly cototient number

Start with the simplest possible case. Write down what Highly cototient 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 Highly cototient 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 Highly cototient 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 Highly cototient number

In research: Highly cototient 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 Highly cototient 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: Highly cototient 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 Highly cototient 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 Highly cototient number in 20 minutes

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

Frequently asked questions

What is Highly cototient number in simple terms?

In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k} than any other integer below k {\displaystyle k} and above 1. Here, ϕ {\displaystyle \p…

Why does Highly cototient 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 Highly cototient 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 Highly cototient number.