Divisibility of Primes (Part Two) — Clement’s Theorem and Twin Primes
Welcome back! In this article, we extend our considerations to a special type of primes called twin primes. A twin prime pair is any pair of prime numbers that have a difference of two. That is, if p and p + 2 are both prime, they are called a twin prime pair. After the distribution of primes, one of the next biggest unsolved problems in prime number theory is that of the twin prime conjecture. A conjecture that hypothesizes the existence of an infinite pairs of twin primes. Consideration of this special type of primes extends all the way back to, in similar fashion to primes, to the ancient Greeks. The interested reader can find a glossary of the first 100,000 twin primes here.
Just as in the case of primes, the twin primes also seem to often elude mathematicians. There are many conjectures regarding the distribution of twin primes or its properties, but only a handful have been proven. That is not to say, however, that we know nothing about twin primes. Twin primes lend themselves to elementary mathematics for some beautiful results.
Firstly, it is fairly easy to show that all twins prime pairs (except the (3,5)) are of the form (6k — 1, 6k+1). Why does this make sense? Well, Since any three consecutive numbers contain a multiple of 3, we know that the number between twin primes is always divisible by 3. Similarly, all primes greater than 2 or odd. This implies that the number between twin primes is always divisible by 2. Combining these results, we see that the “sandwiched” number is always divisible by 6 and therefore, all twin primes are of the form 6k±1.
Another surprising yet powerful result that can be proved using modular arithmetic is that for all twin primes 6k±1, k must end in 0, 2, 3, 5, 7, or 8. A trivial proof follows.
However, as the title suggests, this article is primarily concerned with Clement’s theorem — an extension of Wilson’s theorem. A version of the theorem states the following:
Just as in the case of Wilson’s thereom, this congruence is a necessary and sufficient condition for twin primality. That is, if two numbers satisfy this congruence, they are twin primes and if two numbers are twin primes, they satisfy this congruence. P. A. Clement, in a paper in 1949, provides an elementary proof for this beautiful result:
23–25.
Another characterization of this theorem given by Lin and Zhipeng in their paper from 2004 shows the following:
However, just as in the case of Wilson’s theorem in the last article, both all these representations are of high computational intensity and require really powerful computers to be of any practical use. Even then, owing to the existence of the factorials, it is already very highly inefficient. Why is this? Well, as is well-known, the growth of the factorial function is absurdly quick — even faster than the exponent function. To understand the magnitude of the factorial’s growth, 100! ≈ 9.332622 × 10¹⁵⁷. Yet, mathematical discoveries often have the tendency to unexpectadly be of immense use owing to supplemental discoveries. So, who knows? Maybe someday, these theorems may help solve century-old problems.
Thank you for reading! I hope you have a great day!
