Prime Quadruplet

A Prime Constellation of four successive Primes with minimal distance . 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 quadruplemust be of the form (, , , ). The first few values of which give prime quadruples are, 3, 6, 27, 49, 62, 69, 108, 115, ... (Sloane's A014561), and the first few values of are 5 (the exceptional case),11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, .... The asymptotic Formula for the frequency of prime quadruplesis analogous to that for other Prime Constellations,

References

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. New York: Oxford University Press, 1979.

Forbes, T. ``Prime -tuplets.'' http://www.ltkz.demon.co.uk/ktuplets.htm.

Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-62, 1994.

Roonguthai, W. ``Large Prime Quadruplets.'' http://www.mathsoft.com/asolve/constant/hrdyltl/roonguth.html.

Sloane, N. J. A. Sequence A014561in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

• Sexy Primes
• Seydewitz's Theorem
• Sgn
• Shadow
• Shadowing Theorem
• Shafarevich Conjecture
• Shah Function
• Shah-Wilson Constant
• Shallit Constant
• Shallow Diagonal
• Shanks' Algorithm
• Shanks' Conjecture
• Shannon Entropy
• Shannon Sampling Theorem
• Shape Operator
• Shapiro's Cyclic Sum Constant
• Sharing Problem
• Sharkovsky's Theorem
• Sharpe Ratio
• Sharpe's Differential Equation
• Sheaf (Geometry)
• Sheaf (Topology)
• Shear
• Shear Matrix
• Shephard's Problem