Noncototient

Key takeaways

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

Reference excerpt

In number theory, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then

p q − φ ( p q ) = p q − ( p − 1 ) ( q − 1 ) = p + q − 1 = n − 1. {\displaystyle {\begin{aligned}pq-\varphi (pq)&=pq-(p-1)(q-1)\\&=p+q-1\\&=n-1.\end{aligned}}}

It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1 = 2 − φ(2), 3 = 9 − φ(9), and 5 = 25 − φ(25). For even numbers, it can be shown

2 p q − φ ( 2 p q ) = 2 p q − ( p − 1 ) ( q − 1 ) = p q + p + q − 1 = ( p + 1 ) ( q + 1 ) − 2 {\displaystyle {\begin{aligned}2pq-\varphi (2pq)&=2pq-(p-1)(q-1)\\&=pq+p+q-1\\&=(p+1)(q+1)-2\end{aligned}}}

Thus, all even numbers n such that n + 2 can be written as (p + 1)(q + 1) with p, q primes are cototients. The first few noncototients are

10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, ... (sequence A005278 in the OEIS) The cototient of n are

0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, ... (sequence A051953 in the OEIS) Least k such that the cototient of k is n are (start with n = 0, 0 if no such k exists)

1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, ... (sequence A063507 in the OEIS) Greatest k such that the cototient of k is n are (start with n = 0, 0 if no such k exists)

1, ∞, 4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, ... (sequence A063748 in the OEIS) Number of ks such that k − φ(k) is n are (start with n = 0)

1, ∞, 1, 1, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 4, 4, 3, 0, 4, 1, 4, 3, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, ... (sequence A063740 in the OEIS) Erdős (1913–1996) and Sierpinski (1882–1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family 2 k ⋅ 509203 {\displaystyle 2^{k}\cdot 509203} is an example (See Riesel number). Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).

References Browkin, J.; Schinzel, A. (1995). "On integers not of the form n-φ(n)". Colloq. Math. 68 (1): 55–58. doi:10.4064/cm-68-1-55-58. Zbl 0820.11003. Flammenkamp, A.; Luca, F. (2000). "Infinite families of noncototients". Colloq. Math. 86 (1): 37–41. doi:10.4064/cm-86-1-37-41. Zbl 0965.11003. Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 138–142. ISBN 978-0-387-20860-2. Zbl 1058.11001.

External links Noncototient definition from MathWorld

Worked examples

Example 1 — a first encounter with Noncototient

Start with the simplest possible case. Write down what Noncototient claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Noncototient 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 Noncototient 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 Noncototient

In research: Noncototient appears in science 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 Noncototient 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: Noncototient 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 Noncototient 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 Noncototient in 20 minutes

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

Frequently asked questions

What is Noncototient in simple terms?

In number theory, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m.

Why does Noncototient matter?

Because it connects several science 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 Noncototient?

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 Noncototient.