mathematics

Pernicious number

Pernicious 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 Pernicious number rather than just read about it. In short: In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number. Examples The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime.

Key takeaways

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

Reference excerpt

Examples The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 1012, followed by 6 (1102), 7 (1112) and 9 (10012). The sequence of pernicious numbers begins

Properties No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a Hamming weight of one, and one is not considered to be a prime. On the other hand, every number of the form 2 n + 1 {\displaystyle 2^{n}+1} with n > 1 {\displaystyle n>1} , including every Fermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number. A Mersenne number 2 n − 1 {\displaystyle 2^{n}-1} has a binary representation consisting of n {\displaystyle n} ones, and is pernicious when n {\displaystyle n} is prime. Every Mersenne prime is a Mersenne number for prime n {\displaystyle n} , and is therefore pernicious. By the Euclid–Euler theorem, the even perfect numbers take the form 2 n − 1 ( 2 n − 1 ) {\displaystyle 2^{n-1}(2^{n}-1)} for a Mersenne prime 2 n − 1 {\displaystyle 2^{n}-1} ; the binary representation of such a number consists of a prime number n {\displaystyle n} of ones, followed by n − 1 {\displaystyle n-1} zeros. Therefore, every even perfect number is pernicious.

Related numbers Odious numbers are numbers with an odd number of 1s in their binary expansion (OEIS: A000069). Evil numbers are numbers with an even number of 1s in their binary expansion (OEIS: A001969).

References

Worked examples

Example 1 — a first encounter with Pernicious number

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

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

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

Frequently asked questions

What is Pernicious number in simple terms?

In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number. Examples The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime.

Why does Pernicious 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 Pernicious 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 Pernicious number.