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 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 = 1.
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.
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)