mathematics

Sexy primes

Sexy primes 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 Sexy primes rather than just read about it. In short: In number theory, sexy primes are prime numbers that differ from another prime by 6. For example, the numbers 5 and 11 are a pair of sexy primes, because both are prime and 11 − 5 = 6 {\textstyle 11-5=6} .

Key takeaways

Sexy primes belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sexy primes to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sexy primes from memory before moving on to harder problems.

Reference excerpt

In number theory, sexy primes are prime numbers that differ from another prime by 6. For example, the numbers 5 and 11 are a pair of sexy primes, because both are prime and 11 − 5 = 6 {\textstyle 11-5=6} . The term "sexy prime" is a pun stemming from the Latin word for six: sex. If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.

Sexy prime conjecture The sexy prime conjecture is a weaker analogue of the twin prime conjecture. It asserts that there exist infinitely many pairs of prime numbers that differ by 6, known as sexy primes. Let pn denote the nth prime number, and define

H m = lim inf n → ∞ ( p n + m − p n ) {\displaystyle H_{m}=\liminf _{n\to \infty }(p_{n+m}-p_{n})}

The quantity Hm is the smallest gap that occurs infinitely often between primes separated by m positions in the sequence of prime numbers. In particular, the sexy prime conjecture is equivalent to the statement

H 1 ≤ 6 {\displaystyle H_{1}\leq 6} , which asserts that prime gaps of size at most 6 occur infinitely often. Since every prime gap greater than 2 is even, this is equivalent to the existence of infinitely many sexy prime pairs. In August 2014, the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. Thus, the proof of the sexy prime conjecture is equivalent to the proof of the generalized Elliott-Halberstam conjecture.

Examples The sexy primes (sequences OEIS: A023201 and OEIS: A046117 in OEIS) below 500 are:

(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467). The only sexy primes quintuplet is (5,11,17,23,29).

References

External links Weisstein, Eric W. "Sexy Primes". MathWorld. Grime, James. Brady Haran (ed.). "Sexy Primes (and the only sexy prime quintuplet)". Numberphile. Archived from the original on 23 October 2018.

Worked examples

Example 1 — a first encounter with Sexy primes

Start with the simplest possible case. Write down what Sexy primes 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 Sexy primes 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 Sexy primes 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 Sexy primes

In research: Sexy primes 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 Sexy primes 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: Sexy primes is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classes of prime numbers, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Sexy primes 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 Sexy primes in 20 minutes

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

Frequently asked questions

What is Sexy primes in simple terms?

Why does Sexy primes 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 Sexy primes?

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 Sexy primes.