19 Jul 2012 @ 21:20, by John Grieve
The number system contains symmetries, one class of which I will demonstrate here.
The remainders when a product is divided by its factors are zeros. That is, for example 10 = 0 Mod 5 and 10 = 0 Mod 2. These zeros are important because they create a mirror-image type of symmetry around the product. Thus, if we take the product 15 = 3*5 then in the sequence of numbers before and after 15, there is a mirror-image of the factors 3 & 5.
5 / 7 / 9 / 11 / 13 / 15 / 17/ 19/ 21/ 23 / 25
5 /P / 3/ TP / TP/ 3.5/ TP/ TP/ 3/ P / 5
where P = Prime , TP = Twin Prime and the number underneath is a factor.
From the diagram we can easily see that the order of the factors is reversed.
This interesting fact is true of any product, because the remainders are set to zero, but is particularly relevant to primorial numbers. A primorial is a product of all the primes up to a given prime. So primorial 7 consists of 2*3*5*7=210.
Numbers which are almost primorial are relevant here too. Primorials are important because they have more factors than the numbers around them.
When you multiply the numbers 3,5 and 7 together you get a product which has a large field of mirror-image symmetry around it. Since the numbers 3,5 and 7 create about 70% of composites, it is not surprising that the underlying symmetry of the composites is accompanied in many cases by a corresponding symmetry of the primes and twin primes.
The next prine is 11, and as the primorials get larger, the underlying mirror-image symmetries get richer and more complex.
The fact of these symmetries among composites and primes, may make it easier to demonstrate previously inaccessible propositions about prime numbers.
|
|
