A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible.
   The idea is to generalise the concept of prime twins. More precisely: We first define s(k) to be the smallest number s for which there exist k integers b₁ < b₂ < ... < b_k, b_k − b₁ = s and,
   for every prime q, not all the residues modulo q are represented by b₁, b₂, ..., b_k.    A prime k-tuplet is then defined as a sequence of consecutive primes {p₁, p₂, ..., p_k} such that for every prime q,
   not all the residues modulo q are represented by p₁, p₂, ..., p_k, p_k − p₁ = s(k).
   Observe that the definition might exclude a finite number (for each k) of dense clusters at the beginning of the prime number    sequence − for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet,
   but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.

   The simplest case is s(2) = 2, corresponding to prime twins: {p, p + 2}.
   Next, s(3) = 6 and two types of prime triplets: {p, p + 2, p + 6} and {p, p + 4, p + 6}, followed by s(4) = 8 with just one pattern: {p, p + 2, p + 6, p + 8} of prime quadruplets.
   The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on.
   It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.

   In keeping with similar published lists, I have decided not to accept anything other than true, proven primes. Numbers which have merely passed the Fermat test, a^N = a (mod N),
   will need to be validated. If N − 1 or N + 1 is sufficiently factorized (usually just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will suffice.
   Otherwise the numbers may have to be subjected to a general primality test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and Lenstra (APRT-CLE in UBASIC, for example),
   or one of the elliptic curve primality proving programs: Atkin and Morain's ECPP, or its successor, Franke, Kleinjung, Wirth and Morain's FAST-ECPP, Marcel Martin's Primo or Andreas Enge's CM.

   Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem.
   My favourite (which is not in Ribenboim's book) goes like this: We have

∏_{p prime} 1/(1 − 1/p²) = ∑_{n = 1 to ∞} 1/n² = π²/6.

   But π² is irrational; so the product on the left cannot have a finite number of factors.

   In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x).
   This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:

∫_{u = 0 to x} du/(log u) + error term,

   where the error term is bounded above by A x exp(−B √(log x)) for some constants A and B. With more work (H.-E. Richert, 1967), √(log x) in this last expression can be replaced by (log x)^3/5(log log x)^−1/5.
   The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis − asserts that the error term can be bounded by a function of the form A √x log x.

   G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:

2 C₂ x / (log x)²,

   where C₂ = ∏_{p prime, p > 2} p(p − 2) / (p − 1)² = 0.66016... is known as the twin prime constant.

   V. Brun showed that the sequence of twins is thin enough for the series ∑_{p and p + 2 prime} 1 / p to converge. The twin prime conjecture states that the sum has infinitely many terms.
   The nearest to proving the conjecture is Jing-Run Chen's result that there are infinitely many primes psuch that p + 2 is either prime or the product of two primes [HR73].

   The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k-tuplets of this site are special cases):
   Let b₁, b₂, ..., b_k be k distinct integers. Then the number of groups of primes N + b₁, N + b₂, ..., N + b_k between 2 and x is approximately

H_k C_k ∫_{u = 2 to x} du / (log u)^k,

H_k = ∏_{p prime, p ≤ k} p^{k − 1} (p − v) / (p − 1)^k ∏_{p prime, p > k, p|D} (p − v) / (p − k),

C_k = ∏_{p prime, p > k} p^{k − 1} (p − k) / (p − 1)^k,

   The first product in H_k is over the primes not greater than k, the second is over the primes greater than k which divide D and the product C_k is over all primes greater than k.
   If you put k = 2, b₁ = 0 and b₂ = 2, then v(2) = 1, v(p) = p − 1 for p > 2, H₂ = 2, and C_k = C₂, the twin prime constant given above.

   It is worth pointing out that with modern mathematical software the prime k-tuplet constants C_k can be determined to great accuracy. The way not to do it is to use the defining formula.
   Unless you are very patient, calculating the product over a sufficient number of primes for, say, 20 decimal place accuracy would not be feasible.
   Instead there is a useful transformation originating from the product formula for the Riemann ζ function:

log C_k = − ∑_{n = 2 to ∞} log [ζ(n) ∏_{p prime, p ≤ k} (1 − 1/pⁿ)] / n ∑_d|n μ(n/d) (k^d − k).