mathematics

Prime triplet

Prime triplet 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 Prime triplet rather than just read about it. In short: In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6).

Key takeaways

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

Reference excerpt

In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).

Examples The first prime triplets (sequence A098420 in the OEIS) are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887)

Subpairs of primes A prime triplet contains a single pair of:

Twin primes: (p, p + 2) or (p + 4, p + 6); Cousin primes: (p, p + 4) or (p + 2, p + 6); and Sexy primes: (p, p + 6).

Higher-order versions A prime can be a member of up to three prime triplets - for example, 103 is a member of (97, 101, 103), (101, 103, 107) and (103, 107, 109). When this happens, the five involved primes form a prime quintuplet. A prime quadruplet (p, p + 2, p + 6, p + 8) contains two overlapping prime triplets, (p, p + 2, p + 6) and (p + 2, p + 6, p + 8).

Conjecture on prime triplets Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are (p, p + 2, p + 6) with p = 2072644824759 × 233333 − 1. As of October 2020 the largest known proven prime triplet contains primes with 20008 digits, namely the primes (p, p + 2, p + 6) with p = 4111286921397 × 266420 − 1. The Skewes number for the triplet (p, p + 2, p + 6) is 87613571, and for the triplet (p, p + 4, p + 6) it is 337867.

References

External links Weisstein, Eric W. "Prime Triplet". MathWorld. OEIS sequence A022004 (Initial members of prime triples (p, p+2, p+6)) OEIS sequence A022005 (Initial members of prime triples (p, p+4, p+6))

Worked examples

Example 1 — a first encounter with Prime triplet

Start with the simplest possible case. Write down what Prime triplet 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 Prime triplet 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 Prime triplet 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 Prime triplet

In research: Prime triplet 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 Prime triplet 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: Prime triplet 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 Prime triplet 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 Prime triplet in 20 minutes

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

Frequently asked questions

What is Prime triplet in simple terms?

Why does Prime triplet 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 Prime triplet?

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 Prime triplet.