Is it trite to say that prime numbers are unique? Probably. They are quintessentially unique. Pattern-seeking among prime numbers scattered here and there along the number line has ended many a time in a headache and a sleepless night. Why are they distributed so unevenly along the number line? And why, among the first 30 million numbers, are only 4 of the prime numbers perfect numbers?
Based on the work of Omar E. Pol, this stunning visualization by Jason Davies reveals the interplay of each number’s unique pattern, displayed as a periodic curve, superposed with the unique pattern of every other number. “For each natural number n, we draw a periodic curve starting from the origin, intersecting the x-axis at n and its multiples. The prime numbers are those that have been intersected by only two curves: the prime number itself and one.”
It begs the question: is it the superposition of these different patterns — these unique periodic curves — that causes the seeming irregularity of prime numbers? Something to ponder.
“This pattern cannot merely be a coincidence. A mathematician who finds a pattern of this sort with instinctively ask, ‘Why? What is the reason behind this order?’ Not only will all mathematicians wonder what the reason is, but even more importantly, they will all implicitly believe that whether or not anyone ever finds the reason, there must be a reason for it. Nothing happens ‘by accident’ in the world of mathematics. The existence of a perfect pattern, a regularity that goes on forever, reveals — just as smoke reveals a fire — that something is going on behind the scenes. Mathematicians consider it a sacred goal to seek that thing, uncover it, and bring it out into the open.” — Douglas Hofstadter (I Am A Strange Loop, p. 117)
DH, while ready to admit that it is certainly a pretty image, was quick to point out that it isn’t anything mathematically new. While true, there is a usefulness to this pretty image: it provides a way of determining the primality (even the perfection) of numbers as far out on the number line as you can imagine. An elegant improvement on prime number calculators, I think.