Goldbach's Conjecture - Visualized

Goldbach's Conjecture

"The longest-standing math problem is Goldbach's Conjecture, posed by the Russian mathematician in 1742, according to the Guinness Book of World Records. The theorem states that every even positive integer greater than three is the sum of two prime numbers." -- Zoe Mintz - IBTimes

I was interested to get a feel for this conjecture, where an even number was the sum of two prime numbers. I took a list of the prime numbers up to 32767, and knew I'd have enough prime numbers to cover to 32768.

First, I looked to see if these numbers matched the condition. Using Java, a quickly found that these first few even numbers (the first 16000) matched at least one pair of prime numbers.

Next I wanted to see how constrained these values were. Instead of checking whether two primes existed that added up to N, I checked how many unique pairs of primes matched. Matching more than one pair was a dramatic understatement. By 3000, each even integer could be represented by the sum of at least 20 pairs of primes. That resulted in this chart which I produced in Excel:

This is definitely not an arbitrary chart. Individual values are all over the place, but in general, there are clear comet trails that extend, and a cone-shaped ballistics pattern emerges; definitely not what I had expected. I was expecting something more like the pattern made by a can of spray paint squirted for a second.

In Excel, you can mouse-over a point in the chart, and I was looking at outliers. I noticed that many of the outliers were multiples of 10. Redoing the above chart, I selected out 0 mod 10 (multiples of 10) and the non-0 mod 10.  How does that look? 

Red and Green show the values bifurcate fairly cleanly between two 0-mod-10 paths, and non-0-mod-10 paths -- two of each. Why is that? 

I plotted a section with an overlay of 0-mod-16 and it clusters as well. The number of prime-pairs that can add to an even number appears influenced either by the prime factors themselves, or the demographics of the prime factors. 0-mod-16 is more dissimilar to 0-mod-10 than like, and in terms of the measure based on the number of prime-pair combinations, 

0-mod-16 is less productive on average than 0-mod-10. 0-mod-16 has more prime factors, though fewer different prime factors. If prime factors is a relevant factor in this phenomenon, more different is more important than more total, probably.

I bucketed the data a third way looking at groups of 5 values each, value-div-10 and looked at the max, min, and average counts of ways to combine two primes into one even number, and got this chart:

Within each bucket group, the max values, min values, and average values cluster pretty closely. The minimum value is quite tight. The implication is also that even in groups of 5, max and min values oscillate quite regularly. The trend lines on MAX and AVG are across 100 buckets in this chart, and MIN is across 8 buckets. Times 5, that would be 40 even numbers.

Next Steps

I'd like to redo my program for primes and even numbers much bigger than 32767, selecting a range essentially at random, to verify the pattern continues. I expect that more clustered comet tails will be revealed.

How do the number of combinations of P1 and P2 relate to the number of prime factors in the even number? Multiples of 10 greater than 10 will generally have at least three prime factors, 2, 5, and at least one other number.


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