The "last prime" (aka why the twin prime proof didn't work)

It was axn from Mersenneforum who explained to me why my proof failed. He said "This is where your proof breaks down. Why "must" there be such a "last prime"?" He was quite right (I was relying on the idea that it was impossible for anyb "last prime" to cast out all remaining twin prime candidate pairs.)

Euclid's proof of the infinity of primes basically works because the symmetrical pattern built up by the primes must leave a gap for any finite set. This gap must be filled by a higher prime than the set we were considering (or a multiple of such a prime).

This works because it means that we need a higher prime to fill the gap.

In the case of twin primes, the existence of the gap in the pattern simply means that we need a higher prime, not a higher pair of twin primes.

So the proof would additionally need to show that you can't close all the gaps without moving up through the sieve to ever higher twin primes - otherwise it doesn't work.

I was grateful to him for spotting the problem and explaining it patiently.

So a failure, but I learnt something along the way, if only how difficult this problem is.

11# Twin Prime Symmetry

When I started looking at twin primes I got fascinated by the primorial length symmetrical patterns formed by composite numbers that are multiples of a limited set of factors (eg the pattern for multiples of 5 and 7 is symmetrical over a primorial length pattern of 210 numbers, and so is the pattern for twin prime candidates, pairs of 6N+/-1 numbers for which neither is a multiple of 5 and 7).

It can be hard to picture the symmetries I am talking about due to the length of series involved. Here is an image of the start, finish and centre of the 5/7/11 twin prime symmetry. You can see that the pairs struck out at 11 and 2299, 2101 and 209, and 2057 and 253 are the only pairs cast out by adding 11 to the sieve in this section, and that they are symmetrically arranged (as are all the other pairs cast out by 5 and 7).

Similarly you can see the pairs containing 1067 and 1243, and 1199 and 1111 are cast out by 11 and are symmetrical around the central point between 1151/1153 and 1157/1159.

 

Note that there is a gap of 2 spaces at the centre of the symmetry. Adding 11 to the symmetry has led to 30 pairs being cast out - 15 above the centre and 15 below.

These include the pair 11 and 13 which is of course a twin prime even though it is included in this symmetry.

135 of 385 twin pair candidates remain in this pattern from start to finish and the pattern will repeat from 2310 to 4620.

Fractal Primorial Patterns - a few basics

Lots of people have noticed primorial patterns in the primes. Just a quick note on one way of thinking about these:

Each time we go up through the sieve of Eratosthenes we get a primorial length symmetrical pattern of multiples. This pattern is endlessly repeated.

So, at a very basic level, after we have put 2 and 3 in the sieve we have the pattern

1 2 3  4   5   6
7 8 9 10 11 12

This is symmetrical (with 3 as a centre of symmetry of the first row) and will endlessly repeat.

The same is true after we sieve 5. This leaves a symmetrical pattern of 1, 7, 11, 13, 17, 19, 23, 29, which repeats in each group of 30 numbers.

If we make a sieve of 210 columns for the sieve up to 7, or 2310 columns for the sieve up to 11 we see the same kind of pattern - a symmetry of gaps which repeat on each subsequent iteration.

This process creates a fractal pattern, a overlaid pattern of self-similar elements. But it is not strictly a fractal pattern of primes or composites, because some primes fall within the pattern. It is a fractal pattern in which the gaps indicate "numbers that are not multiples of the primes {2....p}" (including 1 x 2, 1 x 3, 1 x 5 .... 1 x p).

Each time a primorial length pattern is reiterated it is also important to note what happens to each repeating gap (for instance 29, 59, 89, 119, 149, 179, 209). Within the prime pattern, the next prime p in the sieve will only remove 1/p of each particular gap (above you can see that 7 sieves out 119 from this particular set of gaps).

The overall pattern of gaps sieved out by the next prime is also symmetrical. Each pattern is symmetrical around a multiple of the previous primorial (so at the stage 7 is added, the symmetry is around multiples of 30, when 11 is added, the symmetry is around multiples of 210 etc). So numbers sieved in the first half of each iteration of the pattern are also sieved in a mirror pattern in the second half.

My failed twin prime proof relied on showing that we get the same kind of symmetrical pattern of gaps in the layout of twin prime pairs, and that the next prime only removes 2/p of the pairs in each line of gaps. This is correct but it only proves that when we sieve with any finite set of primes, there is an infinite, repeating pattern of "numbers that are not multiples of those primes".

Fractal Patterns in N-Factor Numbers

This is irrelevant to twin primes but it is interesting to observe another way in which the distribution of primes creates a fractal pattern.

The pattern of primes eg 1 x 5, 1 x 7, 1 x 11 etc is also fractally replicated in n-factor numbers. For 2-factor numbers a new pattern (congruent to the overall pattern of primes) starts at each prime  eg 5 x 5, 5 x 7, 5 x 11…  7 x 5, 7 x 7, 7 x 11…. and so on. For 3-factor numbers we have a new repeat of the prime pattern starting from each 2-factor number and so on.

Each iteration of this pattern produces a set of numbers. A set of composite numbers with up to n factors where n is finite cannot define every possible composite number. So the pattern is replicated endlessly, in another fractal pattern, creating composites of n+1, n+2 etc factors.