A prime constellation of four successive primes with minimal distance 
. The term was coined by Paul Stäckel (1892-1919;
Tietze 1965, p. 19). The quadruplet (2, 3, 5, 7) has smaller minimal distance,
but it is an exceptional special case. With the exception of (5, 7, 11, 13), a prime
quadruple must be of the form (
, 
, 
, 
). The first few values of 
which give prime quadruples are 
, 3, 6, 27, 49, 62, 69, 108, 115, ... (OEIS A014561),
and the first few values of 
are 5 (the exceptional case), 11, 101, 191, 821, 1481, 1871,
2081, 3251, 3461, ... (OEIS A007530). The number
of prime quadruplets with largest member less than 
, 
, ..., are 1, 2, 5, 12, 38, 166, 899, 4768, ... (OEIS A050258; Nicely 1999).
The asymptotic formula for the frequency of prime quadruples is analogous to that for other prime constellations,
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(1)
|
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(2)
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where 
(OEIS A061642) is the Hardy-Littlewood constant
for prime quadruplets.
Roonguthai found the large prime quadruplets with
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(3)
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
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(9)
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Forbes found the large quadruplet with
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(10)
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-tuplets." 
of the Prime Quadruplets." Submitted to
Math. Comput.