Let:f(x,y)=∣6xy+x+y∣
where 𝑥,𝑦 ∈ 𝑍∖{0} (i.e., both are non-zero integers, so may be positive or negative).
Define the set:
𝐾composite = {𝑓(𝑥,𝑦): 𝑥≠0, 𝑦≠0}
Then: A positive integer 𝑘 is the index of a twin prime pair (6𝑘−1,6𝑘+1) if and only if:
𝑘∉𝐾composite
Therefore, the Twin Prime Conjecture is true if and only if:
𝑍+∖𝐾composite is infinite
In plain language:
There are infinitely many twin primes if and only if there are infinitely many positive integers 𝑘 that cannot be written in the form ∣6𝑥𝑦+𝑥+𝑦∣ for any non-zero integers 𝑥,𝑦.