mathematics

Primeval number

Primeval 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 Primeval number rather than just read about it. In short: In recreational number theory, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith.

Key takeaways

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

Reference excerpt

In recreational number theory, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith. The first few primeval numbers are

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ... (sequence A072857 in the OEIS) The number of primes that can be obtained from the primeval numbers is

0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, ... (sequence A076497 in the OEIS) The largest number of primes that can be obtained from a primeval number with n digits is

1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ... (sequence A076730 in the OEIS) The smallest n-digit number to achieve this number of primes is

2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ... (sequence A134596 in the OEIS) Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:

2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379, ... (sequence A119535 in the OEIS) The following table shows the first seven primeval numbers with the obtainable primes and the number of them.

Base 12 In base 12, the primeval numbers are: (using inverted two and three for ten and eleven, respectively)

1, 2, 13, 15, 57, 115, 117, 125, 135, 157, 1017, 1057, 1157, 1257, 125Ɛ, 157Ɛ, 167Ɛ, ... The number of primes that can be obtained from the primeval numbers is: (written in base 10)

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 20, 23, 27, 29, 33, 35, ...

Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31.

See also Permutable prime Truncatable prime

External links Chris Caldwell, The Prime Glossary: Primeval number at The Prime Pages Mike Keith, Integers Containing Many Embedded Primes

Worked examples

Example 1 — a first encounter with Primeval number

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

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

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

Frequently asked questions

What is Primeval number in simple terms?

Why does Primeval 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 Primeval 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 Primeval number.