mathematics

Untouchable number

Untouchable 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 Untouchable number rather than just read about it. In short: In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function.

Key takeaways

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

Reference excerpt

In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.

Examples

The number 4 is not untouchable, as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2). The number 6 is not untouchable, as it is equal to the sum of the proper divisors of 6 itself: 1 + 2 + 3 = 6. The first few untouchable numbers are

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence A005114 in the OEIS).

Properties

The number 5 is believed to be the only odd untouchable number, but this has not been proven. It would follow from a slightly stronger version of the Goldbach conjecture, since the sum of the proper divisors of pq (with p, q distinct primes) is 1 + p + q. Thus, if a number n can be written as a sum of two distinct primes, then n + 1 is not an untouchable number. It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 7 is an untouchable number, and 1 = σ ( 2 ) − 2 {\displaystyle 1=\sigma (2)-2} , 3 = σ ( 4 ) − 4 {\displaystyle 3=\sigma (4)-4} , 7 = σ ( 8 ) − 8 {\displaystyle 7=\sigma (8)-8} , so only 5 can be an odd untouchable number. Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers (since except 2, all even numbers are composite). No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors. Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, none of the Mersenne numbers are untouchable, since Mn = 2n − 1 is equal to the sum of the proper divisors of 2n. No untouchable number is 1 more than a prime number, since if p is prime, then the sum of the proper divisors of p2 is p + 1. Also, no untouchable number is 3 more than a prime number, except 5, since if p is an odd prime then the sum of the proper divisors of 2p is p + 3.

Infinitude There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős. According to Chen and Zhao, their natural density is at least d > 0.06.

See also Aliquot sequence Nontotient Noncototient Weird number

References

Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B10.

External links OEIS sequence A070015 (Least m such that sum of aliquot parts of m equals n or 0 if no such number exists)

Worked examples

Example 1 — a first encounter with Untouchable number

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

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

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

Frequently asked questions

What is Untouchable number in simple terms?

Why does Untouchable 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 Untouchable 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 Untouchable number.