Prime Number Fusion

This page is a demonstration of a prime number construction method. The basic idea is to treat the first prime number (2) as the ultimate building block for all other primes. The program follows a simple set of rules and shows the number of steps required to make all prime numbers up to p32 (127). The rules used to construct the primes are as follows;

The Postulates

We assume an unlimited supply of p1's [prime numbers 2]

We assume that two prime numbers may fuse to become the next prime plus a remainder if the prime gap is larger than (2).

We postulate that there are no tri-fusion events, ie. fusion only occurs between two primes.

The Method

From two p1's we construct p2 as follows: (p1 + p1) --> p2 [2 + 2] --> 3

With p2 we are able to construct p3: (p2 + p1) = p3 [3 + 2] --> 5

With p3 we are able to construct p4: (p3 + p1) = p4 [5 + 2] --> 7

Up to this point it is trivial because the prime number gaps equal p1 (2).

The next prime number gap between p4 and p5 is (4) and can not be completed with a p1 (2), ergo one must first first construct another p3 (5) before this process can take place

With p4 an p3 we are able to construct p5: (p4 + p3) --> p5 plus a remainder p2 [7 + 5] --> [11 + 3]

...etc

The Inspiration

The idea for this function comes from the nuclear fusion of elements in stars, where the elementary building block is hydrogen which is fused into the heavier elements. One can imagine the proton as p1 or the real number (2) and Deuterium as p2 or the real number 3, and Tritium as p3 or the real number (5). In a star as in my program, elements can not be created until their sub components have been made available.

Interestingly we see from the table below that the number of steps required to build a prime number is equal to sum of the ordinals, i.e. after 149 steps we have the following stock;

1 x p1

1 x p2

1 x p5

1 x p7

3 x p31

1 x p32

Which as we can check sums to 149.

One may speculate that composite numbers are simply steps involved in making the primes. It would be interesting to see if this sort of calculation with primes could be extended to a consistent mathematical theory.

Further Speculation

Everyone is familiar with the standard model of cosmology and how the Universe presumably began with a big bang. Although a fair amount of evidence has been constructed for the big bang theory, it does not convincingly solve the "hokus pokus" (magic) of how everything we observe originated from a singularity, this part of the theory is unconvincing.

I have proposed an alternative theory in my paper titled Ground Potential Theory [1] where time and space began with a single hydrogen atom fusing with another and becoming Deuterium, and then fusing with another to become tritium, then Helium, Berylium, Boron and so on, very much like this proposed prime number fusion.

For those of you who are familiar with the physics of fusion, you will know that each fusion event releases energy according to E=mc^2 equation and consequently loses potential. Yet, according to my calculations in Ground Potential [1], our Universe is still young , we have only reached the potential of iron. One could say that we are in the iron age, i.e. earths ground potential is almost at the same potential as the mass per nucleon of iron or Fe56.

This means, all the fusion which has taken place since the beginning of time has on average not taken us further than row 56 in the table below, just a few rows more than shown. The reason for this is the energy released with each reaction. I show in Ground Potential Theory how a change in potential is related to change in velocity; ∆v = c[∆U/Ø], where ∆v is the difference in velocity, c is the speed of light, ∆U is the difference in potential and Ø is absolute potential. It is not strictly necessary to understand why this is the case, just to understand how fusion produces heat and that heat is velocity. We can understand why fusion is such a difficult process to achieve in the real world because the more successful a fusion process is, the hotter it gets and the faster relative movements between atoms are. Contrary to popular misbelief, a hot plasma is not the best way to fuse atoms. Think of a passenger on a platform trying to board on a train whizzing past at 100 mph, it's impossible, the same goes for atoms, they generally have to travel at the same speed and the same direction in order to fuse. Fortunately for us, stars produce a lot of heat, which in turn slows down the fusion process to a mere trickle that lasts for billions of years.

Without going too far off topic, look at the Ground Potential scenario and see how it makes perfect sense for prime numbers to pop into existence only as and when required to describe the current world. Imagine an infant universe with only 3 particles, of what possible use would the number 5 be ?

There are many questions and not enough answers (P vs NP problem) for which I think Ground Potential Theory may also have a solution. Theoretically we could know everything about the past and nothing about the future, so what ground Potential theory tells us is, how much we could know of all there is to know, it can actually give us a precise answer. The theory predicts that our current knowledge of the world is 0.852 % of all there is to know, so don't give up yet, 99% of everything has not yet been discovered.

With that I shall leave you to speculate on how to collect that millennium price, haha :)

The following code was used to construct the table

This may not be the optimum way to code this, as it is easy to make a mistake, so if someone smarter than me has a better suggestion I would love to learn about it. Ideally we would like a program that can calculate the steps for any known prime.

This page is written in php and anyone who wants a copy of the program can contact me.

for ($i = 1; $i < 150; $i++ ) {
if ($p1 >= 2 ) { $p1--; $p1--; $p2++ ;}	# 3-2 gap 1
if ($p2 >= 1 && $p1 >= 1) { $p2--; $p1--; $p3++; }	# 5-3 gap 2
if ($p3 >= 1 && $p1 >= 1) { $p3--; $p1--; $p4++; }	# 7-5 gap 2
if ($p4 >= 1 && $p3 >= 1) { $p4--; $p3--; $p5++; $p2++; }	# 11-7 gap 4
if ($p5 >= 1 && $p1 >= 1) { $p5--; $p1--; $p6++; }	# 13-11 gap 2
if ($p6 >= 1 && $p3 >= 1) { $p6--; $p3--; $p7++; $p2++;}	# 17-13 gap 4
if ($p7 >= 1 && $p1 >= 1) { $p7--; $p1--; $p8++; }	# 19-17 gap 2
if ($p8 >= 1 && $p3 >= 1) { $p8--; $p3--; $p9++; $p2++;}	# 23-19 gap 4
if ($p9 >= 1 && $p4 >= 1) { $p9--; $p4--; $p10++; $p3++; }	# 29-23 gap 6
if ($p10 >= 1 && $p1 >= 1) { $p10--; $p1--; $p11++; }	# 31-29 gap 2
if ($p11 >= 1 && $p4 >= 1) { $p11--; $p4--; $p12++; $p3++;}	# 37-31 gap 6
if ($p12 >= 1 && $p3 >= 1) { $p12--; $p3--; $p13++; $p2++;}	# 41-37 gap 4
if ($p13 >= 1 && $p1 >= 1) { $p13--; $p1--; $p14++;}	# 43-41 gap 2
if ($p14 >= 1 && $p3 >= 1) { $p14--; $p3--; $p15++; $p2++; }	# 47-43 gap 4
if ($p15 >= 1 && $p4 >= 1) { $p15--; $p4--; $p16++; $p3++;}	# 53-47 gap 6
if ($p16 >= 1 && $p4 >= 1) { $p16--; $p4--; $p17++; $p3++;}	# 59-53 gap 6
if ($p17 >= 1 && $p1 >= 1) { $p17--; $p1--; $p18++;}	# 61-59 gap 2
if ($p18 >= 1 && $p4 >= 1) { $p18--; $p4--; $p19++; $p3++;}	# 67-61 gap 6
if ($p19 >= 1 && $p3 >= 1) { $p19--; $p3--; $p20++; $p2++;}	# 71-67 gap 4
if ($p20 >= 1 && $p1 >= 1) { $p20--; $p1--; $p21++;}	# 73-71 gap 2
if ($p21 >= 1 && $p4 >= 1) { $p21--; $p4--; $p22++; $p3++;}	# 79-73 gap 6
if ($p22 >= 1 && $p3 >= 1) { $p22--; $p3--; $p23++; $p2++;}	# 83-79 gap 4
if ($p23 >= 1 && $p4 >= 1) { $p23--; $p4--; $p24++; $p3++;}	# 89-83 gap 6
if ($p24 >= 1 && $p5 >= 1) { $p24--; $p5--; $p25++; $p4++;}	# 97-89 gap 8
if ($p25 >= 1 && $p3 >= 1) { $p25--; $p3--; $p26++; $p2++;}	# 101-97 gap 4
if ($p26 >= 1 && $p1 >= 1) { $p26--; $p1--; $p27++;}	# 103-101 gap 2
if ($p27 >= 1 && $p3 >= 1) { $p27--; $p3--; $p28++; $p2++;}	# 107-103 gap 4
if ($p28 >= 1 && $p1 >= 1) { $p28--; $p1--; $p29++;}	# 109-107 gap 2
if ($p29 >= 1 && $p3 >= 1) { $p29--; $p3--; $p30++; $p2++;}	# 113-109 gap 4
if ($p30 >= 1 && $p7 >= 1) { $p30--; $p7--; $p31++; $p6++;}	# 127-113 gap 14
if ($p31 >= 1 && $p3 >= 1) { $p31--; $p3--; $p32++; $p2++;} }	# 131-127 gap 4

Goldbachs Conjecture

The unproven Goldbachs conjecture states that every whole even number is the sum of two primes, and as we can see from my hypothesis above, whole numbers being the counting steps in prime number fusion and each event is limited to two primes, it is clear that all whole numbers must be the sum of two primes.

I am not a mathematician and I do not know if this could be called a rigorous proof, but it is certainly consistent with the hypothesis.

© Steven Sesselmann
Researchgate Profile
2020-10-11

[Edited 2020-10-12 by Steven] - Formula changed now considers "if (Pn <= 1)

[Edited 2020-10-15 by Steven] - Found several mistakes in the table

[Edited 2020-10-16 by Steven] - Added text under Speculation