mathematics

Super-Poulet number

Super-Poulet 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 Super-Poulet number rather than just read about it. In short: In number theory, a super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor d {\displaystyle d} divides 2 d − 2 {\displaystyle 2^{d}-2} . For example, 341 is a super-Poulet number: it has positive divisors (1, 11, 31, 341), and we have: (211 − 2) / 11 = 2046 / 11 = 186 (231 − 2) / 31 = 2147483646 / 31 = 69273666 (2341 − 2) / 341 = 13136332798696798888899954724741608669335164206654835981…

Key takeaways

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

Reference excerpt

In number theory, a super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor d {\displaystyle d} divides 2 d − 2 {\displaystyle 2^{d}-2} . For example, 341 is a super-Poulet number: it has positive divisors (1, 11, 31, 341), and we have:

(211 − 2) / 11 = 2046 / 11 = 186 (231 − 2) / 31 = 2147483646 / 31 = 69273666 (2341 − 2) / 341 = 13136332798696798888899954724741608669335164206654835981818117894215788100763407304286671514789484550 When Φ n ( 2 ) g c d ( n , Φ n ( 2 ) ) {\displaystyle {\frac {\Phi _{n}(2)}{gcd(n,\Phi _{n}(2))}}} is not prime, then it and every divisor of it are a pseudoprime to base 2, and a super-Poulet number. The super-Poulet numbers below 10,000 are (sequence A050217 in the OEIS):

Super-Poulet numbers with 3 or more distinct prime divisors It is relatively easy to get super-Poulet numbers with 3 distinct prime divisors. If you find three Poulet numbers with three common prime factors, you get a super-Poulet number, as you built the product of the three prime factors. Example: 2701 = 37 * 73 is a Poulet number, 4033 = 37 * 109 is a Poulet number, 7957 = 73 * 109 is a Poulet number; so 294409 = 37 * 73 * 109 is a Poulet number too. Super-Poulet numbers with up to 7 distinct prime factors you can get with the following numbers:

{ 103, 307, 2143, 2857, 6529, 11119, 131071 } { 709, 2833, 3541, 12037, 31153, 174877, 184081 } { 1861, 5581, 11161, 26041, 37201, 87421, 102301 } { 6421, 12841, 51361, 57781, 115561, 192601, 205441 } For example, 1118863200025063181061994266818401 = 6421 * 12841 * 51361 * 57781 * 115561 * 192601 * 205441 is a super-Poulet number with 7 distinct prime factors and 120 Poulet numbers.

External links Weisstein, Eric W. "Super-Poulet number". MathWorld. Numericana

Worked examples

Example 1 — a first encounter with Super-Poulet number

Start with the simplest possible case. Write down what Super-Poulet 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 Super-Poulet 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 Super-Poulet 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 Super-Poulet number

In research: Super-Poulet 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 Super-Poulet 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: Super-Poulet 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 Super-Poulet 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 Super-Poulet number in 20 minutes

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

Frequently asked questions

What is Super-Poulet number in simple terms?

Why does Super-Poulet 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 Super-Poulet 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 Super-Poulet number.