In mathematics, a k-tuple is an ordered set of k values (a vector of degree k):
(a1, a2, ..., ak).
When working in the context of prime numbers a k-tuple usually refers to a k-tuple of distinct positive integers which are used as the constant terms in a k-tuple of linear polynomials:
(x + a1, x + a2, ..., x + ak).
The central questions then are: How often are these polynomial simultaneously prime? For which integer values of x are they all prime?
The twin prime conjecture is equivalent to conjecturing the translates of the 2-tuple (0, 2) (the values of the pair of polynomials (x, x+2)) are simultaneously prime values infinitely often.
What about the 2-tuple (0, 1)? Always either x or x+1 is even, so this 2-tuple takes on simultaneous prime values only when x = 2.
Another example is the 3-tuple (0, 2, 4), this one also can only take on simultaneous prime once, because 3 must always divide one of the three terms (x, x+2, x+4). This means one of these three terms must be 3. Look at this k-tuple modulo 3 and the problem is apparent: (0, 2, 1)--this is a complete residue system for the prime 3.
This is the only thing (as far as in known) that can stop a k-tuple from yielding simultaneous prime infinitely often! For this reason we say that a k-tuple is admissible if it does not include the complete residue system of any prime.
The prime k-tuple conjecture states that each admissible k-tuple takes on simultaneous prime values infinitely often.
As a final example, consider the 5-tuple (5, 7, 11, 13, 17). (Since we are interested in the translates of this tuple, we could equally well just consider (0, 2, 6, 8, 12).) If this contains the complete residue system of any prime, it must be a prime less than or equal to 5 (because there are just 5 terms). Modulo 2 the original tuple is (1, 1, 1, 1, 1), no problem there. Modulo 3 this is (2, 1, 2, 1, 2), again no problem. Modulo 5 we have (0, 2, 1, 3, 2), this omits 4, so again no problem. (5, 7, 11, 13, 17) is admissible.
Related pages (outside of this work)
- A page with Java Script to find admissible k-tuples
- P. Erdös and H. Riesel, "On admissible constellations of consecutive primes," BIT, 28:3 (1988) 391--396. MR 90b:11145
- T. Forbes, "Prime clusters and Cunningham chains," Math. Comp., 68:228 (1999) 1739--1747. MR 99m:11007
- Guy94 (A9)
- R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994. New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, 1995. New York, NY, pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
- Riesel94 (Chapter Three)
- H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics Vol, 126, Birkhäuser Boston, Boston, MA, 1994. ISBN 0-8176-3743-5. MR 95h:11142 [An excellent reference for those who want to start to program some of these algorithms. Code is provided in Pascal. Previous edition was vol. 57, 1985.]