Philosophy of Number: The Shadows of Prime Numbers

Prime numbers are the elemental building blocks that determine the composite numbers, and they famously divide the number line into random increments in a fashion that has baffled mathematicians for centuries. There is an interesting related phenomenon, however, that we can understand. You can categorize sets of multiple primes within the boundaries of two smaller consecutive primes squared, thereby projecting these smaller primes onto larger segments of numbers farther down the number line in a meaningful way. I am calling this effect a shadow. Take all of the primes between 25 and 49 as an example. 29, 31, 37, 41, 43, and 47. Their pattern may not be known, but these are the gaps in the unique shadow produced by projecting 5 squared and 7 squared in which all of the composite numbers in between are uniquely generated from all of the preceding primes up to and including the smaller of the two primes we are squaring to produce the entire space of the shadow. In the case of the shadow between 25 and 49, every composite number can be produced by the formulas 2x, 3x, and 5x with x being all positive integers extending out to infinity. The second we reach 49, we need to use a higher layer of composite numbers to begin producing some of the composites which appear. So, we add a 7x to our list of composite numbers by smallest factor, but we never need to do this until we reach the square of the next prime number at which point the previous shadow changes to a unique new shadow. We can generate all of the composite numbers from 49 up to 121 with the formula layers so far introduced including 7x, but when we get to 121 (11 squared) we need to begin including 11x in our smallest prime factors list to cover all of the composite numbers in the shadow. So the squares of prime numbers present us with real boundaries that extend up the number line, covering larger increments that each contain more prime numbers. By looking at just the composite numbers, we simplify the primes to the gaps between these numbers, the light in the shadows, with these shadow numbers generated from much simpler elements. In this way, the numbers can be thought of as unfolding in groups of numbers which are then projected to the unfolding of larger groups of numbers. These groups are bounded by the squares of each consecutive prime, and their makeup is foreshadowed much earlier than when they appear.