Perfect totient number

Key takeaways

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

Reference excerpt

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

Examples For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number. The first few perfect totient numbers are

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... (sequence A082897 in the OEIS).

Notation In symbols, one writes

φ i ( n ) = { φ ( n ) , if i = 1 φ ( φ i − 1 ( n ) ) , if i ≥ 2 {\displaystyle \varphi ^{i}(n)={\begin{cases}\varphi (n),&{\text{ if }}i=1\\\varphi (\varphi ^{i-1}(n)),&{\text{ if }}i\geq 2\end{cases}}}

for the iterated totient function. Then if c is the integer such that

φ c ( n ) = 2 , {\displaystyle \displaystyle \varphi ^{c}(n)=2,}

one has that n is a perfect totient number if

n = ∑ i = 1 c + 1 φ i ( n ) . {\displaystyle n=\sum _{i=1}^{c+1}\varphi ^{i}(n).}

Multiples and powers of three It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that

φ ( 3 k ) = φ ( 2 × 3 k ) = 2 × 3 k − 1 . {\displaystyle \displaystyle \varphi (3^{k})=\varphi (2\times 3^{k})=2\times 3^{k-1}.}

Venkataraman (1975) found another family of perfect totient numbers: if p = 4 × 3k + 1 is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are

0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... (sequence A005537 in the OEIS). More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3kp where p is prime and k > 3.

References Pérez-Cacho Villaverde, Laureano (1939). "Sobre la suma de indicadores de ordenes sucesivos". Revista Matematica Hispano-Americana. 5 (3): 45–50. Guy, Richard K. (2004). Unsolved Problems in Number Theory. New York: Springer-Verlag. p. §B41. ISBN 0-387-20860-7. Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003). "On perfect totient numbers" (PDF). Journal of Integer Sequences. 6 (4): 03.4.5. Bibcode:2003JIntS...6...45I. MR 2051959. Archived from the original (PDF) on 2017-08-12. Retrieved 2007-02-07. Luca, Florian (2006). "On the distribution of perfect totients" (PDF). Journal of Integer Sequences. 9 (4): 06.4.4. Bibcode:2006JIntS...9...44L. MR 2247943. Retrieved 2007-02-07. Mohan, A. L.; Suryanarayana, D. (1982). "Perfect totient numbers". Number theory (Mysore, 1981). Lecture Notes in Mathematics, vol. 938, Springer-Verlag. pp. 101–105. MR 0665442. Venkataraman, T. (1975). "Perfect totient number". The Mathematics Student. 43: 178. MR 0447089. Hyvärinen, Tuukka (2015). "Täydelliset totienttiluvut". Tampere: Tampereen yliopisto. This article incorporates material from Perfect Totient Number on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Worked examples

Example 1 — a first encounter with Perfect totient number

Start with the simplest possible case. Write down what Perfect totient 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 Perfect totient 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 Perfect totient 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 Perfect totient number

In research: Perfect totient 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 Perfect totient 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: Perfect totient 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 Perfect totient 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.

How to study Perfect totient number in 20 minutes

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

Frequently asked questions

What is Perfect totient number in simple terms?

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers…

Why does Perfect totient 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 Perfect totient 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 Perfect totient number.