Prime quadruplet

Key takeaways

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

Reference excerpt

In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form {p, p + 2, p + 6, p + 8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.

Prime quadruplets The first eight prime quadruplets are: {5, 7, 11, 13}, {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, {1481, 1483, 1487, 1489}, {1871, 1873, 1877, 1879}, {2081, 2083, 2087, 2089} (sequence A007530 in the OEIS) All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. This structure is necessary to ensure that none of the four primes are divisible by 2, 3 or 5. The first prime of all such quadruplets ends with the digit 1 and the last prime ends with the digit 9, in base 10. Thus a prime quadruplet of this form is called a prime decade. A prime quadruplet can be described as a consecutive pair of twin primes, two overlapping sets of prime triplets, or two intermixed pairs of sexy primes. These "quad" primes can also form the core of prime quintuplets and prime sextuplets when adding or subtracting 8 from their centers yields a prime. All prime decades starting above 5 have centers of the form 210n + 15, 210n + 105, or 210n + 195, since the centers must be −1, 0, or 1, modulo 7. The +15 form may also give rise to a (high) prime quintuplet; the +195 form can also give rise to a (low) quintuplet; while the +105 form can yield both types of quintuplets and possibly prime sextuplets. It is no accident that each prime in a prime decade is displaced from its center by a power of 2, since all centers are odd and divisible by both 3 and 5. It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. However, it is conjectured that there are infinitely many prime quadruplets. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is

1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in the OEIS). It is conjectured that the asymptotic formula for the number of prime quadruplets below n is C x ( ln ⁡ x ) 4 {\displaystyle C{\frac {x}{(\ln x)^{4}}}} , where C ∼ 4.151180863 {\displaystyle C\sim 4.151180863} (sequence A061642 in the OEIS). As of February 2019 the largest known prime quadruplet has 10132 digits. It starts with p = 667674063382677 × 233608 − 1, found by Peter Kaiser. The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:

B 4 = ( 1 5 + 1 7 + 1 11 + 1 13 ) + ( 1 11 + 1 13 + 1 17 + 1 19 ) + ( 1 101 + 1 103 + 1 107 + 1 109 ) + ⋯ {\displaystyle B_{4}=\left({\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{101}}+{\frac {1}{103}}+{\frac {1}{107}}+{\frac {1}{109}}\right)+\cdots }

with value:

B4 = 0.87058 83800 ± 0.00000 00005. This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4. The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed. Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {p, p + 2, p + 6, p + 8} and {q, q + 2, q + 6, q + 8} is q − p = 30. The first occurrences of this are for p = 1006301, 2594951, 3919211, 9600551, 10531061, ... (OEIS: A059925). The Skewes number for prime quadruplets {p, p + 2, p + 6, p + 8} is 1172531 (Tóth (2019)).

Prime quintuplets If {p, p + 2, p + 6, p + 8} is a prime quadruplet and p − 4 or p + 12 is also prime, then the five primes form a prime quintuplet which is the closest admissible constellation of five primes. The first few prime quintuplets with p + 12 are:

{5, 7, 11, 13, 17}, {11, 13, 17, 19, 23}, {101, 103, 107, 109, 113}, {1481, 1483, 1487, 1489, 1493}, {16061, 16063, 16067, 16069, 16073}, {19421, 19423, 19427, 19429, 19433}, {21011, 21013, 21017, 21019, 21023}, {22271, 22273, 22277, 22279, 22283}, {43781, 43783, 43787, 43789, 43793}, {55331, 55333, 55337, 55339, 55343} … OEIS: A022006. The first prime quintuplets with p − 4 are:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Prime quadruplet

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

In research: Prime quadruplet 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 quadruplet 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 quadruplet 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 quadruplet 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.

How to study Prime quadruplet in 20 minutes

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

Frequently asked questions

What is Prime quadruplet in simple terms?

In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form {p, p + 2, p + 6, p + 8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.

Why does Prime quadruplet 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 quadruplet?

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 quadruplet.