Sunday, 22 January 2012

Composite numbers seen as the difference between two squares Part 2

OK, so next we want to go from looking for composite numbers adjacent to multiples of 12 to looking for composite numbers adjacent to 24. This is fairly easily done. The square of any number that is 6 (mod 12) will be 12 (mod 24). The square of any 6N+/-1 number is 1 (mod 24), so the difference between the 2 will always be 11 or 13 (mod 24).

So from the previous chart, we can eliminate all the columns for numbers that are 6 (mod 12), in other words 6, 18, 30. And we can also eliminate every second entry in the columns for numbers that = 1 or 5 (mod 6) - so we eliminate 17^2 - 6^2, but not 17^2 - 12^2 and so on. The red numbers here are the eliminated ones.

This eliminates half of the entire pattern (just as we eliminated half of the cells, the orange ones when we moved from composite numbers adjacent to 6 to those adjacent to 12).

At this point I want to use a different style of array to demonstrate the patterns in these numbers.

In this array we unify columns so that we simply subtract the squares of 6N+/-1 numbers from the squares of multiples of 12. This produces all composite numbers that are 1 more or one less than a multiple of 24. 6N+1 numbers are shown as negative numbers, but the important thing is the numerical value, not the sign (because 17^2 - 12 ^2 has the same numerical value as 12^2 - 17^2).

After the column showing the composite numbers generated this way, the next column indicates which multiple of 24 the composite number is adjacent to. For instance 12^2 - 5^2 = 95, which is adjacent to 96 so the cell next to 24 gives the value of 4.

The grey numbers on the diagonal are cases where the difference between N and M is 1 and thus do not indicate a composite number unless the same value appears elsewhere.

The cells coloured pink are all the cells where we have a composite adjacent to a multiple of 24 from 1 to 12, in other words 288 or lower. We will see this basic pattern recur in later diagrams so it is important to understand why this is the only region in which these numbers can be found. I'll look at this in more detail in a later post.

If each number from 1 to 12 were present it would show that there are no twin primes adjacent to the first 12 multiples of 24. In fact there is no instance of 3, 8, or 11, which is what we would expect given that 71/73, 191/193 and 239/241 are twins.

It's also useful to note the reason why these pairs are not present in the pattern. There are 12 cells coloured pink. If each of these were a different numerical value it would be enough to eliminate every pair. However 5, 6 and 9 appear twice in the pattern, once for a 6N+1 number and once for a 6N-1 number. This means there aren't enough composites in this pattern to cover every pair adjacent to 24 up to 288.

(The other reason why a number will appear more than once is where the composite number has 2 or more distinct factorisations. For instance in the next pattern we will see several instances of the number 385. This is because it is 5 x 7 x 11 and thus can be expressed three ways:

35 x 11 = 23^2 - 12 ^2
55 x 7 = 31^2 - 24^2
5 x 77 = 41^2 - 36^2)

Part Three

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