B. A proof.
B.1. An intuitive approach: The Goldbach Conjecture visualised.
An intuitive demonstration of the Goldbach Conjecture, saying that “Each even number that is bigger than 2 can be written as a sum of two prime numbers.”
This demonstration proceeds by the means of, firstly, a ‘reductio ad absurdum’ and, secondly, an auxiliary statement which can be proved easily by the means of a statement by Erdös, and which says that, concerning each even number E that is bigger than 8, it holds that: "The product of all prime factors pi , so that 0<pi <or= (E:2), is always bigger than E itself." Again: as a matter of fact, the number 1 is not been seen as a prime number here.
The following demonstration arose from the intuition that the Goldbach problem couldn't be resolved but on the condition that the division of the numbers both in terms and in factors could be placed in one image simultaneously. Out of this intuition, the idea arose to work by means of representations of numbers which should make visible both mentioned divisions simultaneously. The original prove has an intuitive character. The elimination of informal aspects by the responding of all possible objections, carried us to a formal, algebraic prove, which will be presented later on.
In this intuitive demonstration we will proceed by the means of an example. We do so to make possible a good understanding of the demonstration. The general approach follows at the end of this exposition.
An intuitive approach.
Goldbach says that each even number bigger than 2, can be written as being the sum of two prime numbers.
Let us take an arbitrary number, e.g. the number 8. The mentioned thesis only holds in the case of an E so that E>8, yet this is no objection concerning our exposition. For we want to keep our representation as simple as possible, we must ask the reader to have some patience: the general approach will follow later on.
We now represent the number 8 as follows:
The reason why, from now on, we will represent the numbers in the way practised here, must be clear: we will have to be able to approach each number as being a unity that, in a specific number of ways, can be composed out of different terms at the one hand, and out of different factors at the other hand. This is because the Goldbach-problem concerns a well-specified relationship between, at the one hand, the terms of even numbers and, at the other hand, their divisors. In this way, our manner of representation allows us to observe how the number 8, represented by a fragment with a length of 8, is been composed out of the terms 2 and 6, for we can add mutually these fragments of respectively length 2 and 6, and, at the same time, we can see how the number 8 is being composed out of divisors, e.g. the divisors 2 and 4, specifically as we can see how the product of the divisors 2 and 4 generates 8. In order to be able to observe this well, we will use 'waves'. We first of all must remark that, by this terminology, we do not aim the physical concept of 'waves', as one should normally think: we just use the specific representation of waves for mere didactic reasons. In this way, e.g., this specific representation of the number 8 will show us that 8 contains the factor 2 (in other terms: the number 8 has the number 2 as a divisor), because the 'wave of 2' crosses the horizontal axis at 8. In general, the representation by means of 'waves' shows us how each number is a multiple of those prime numbers which have waves crossing the horizontal axis at the position of that very number.
We know that each even number has a natural number as its half. That half can be an even number, an odd number, a prime number or a compound number.
In our example with the arbitrary chosen even number E=8, the half of that number 8 equals 4.
We now indicate the number 4, being the half of the number E=8, on our representation of the number 8, and we do so by the drawing of a dotted perpendicular line on the axis that carries our representation of the number 8, throughout the 'point 4', as follows:
We now consider all prime numbers that either are smaller than the half of our even number, or equal this half. So, in our example, we consider all prime numbers pi , so that 0<pi<or=4 , being the numbers 2 and 3 in our example. We now indicate these prime numbers by means of the character "P" on our representation of the number E=8, as follows:
Remark: factually, we do not need the prime number 2, because the 'counterpart' of 2 (namely: the specific prime number that has to be added up with 2 in order to get a sum resulting in the even number E that is in question), will be odd, while the sum of an even number (in casu the number 2) and an odd one (for the 'counterpart' of 2 never can be even again because 2 is the only even prime number) can never generate an even number.
Now, due to the Goldbach Conjecture, it must hold for each even number E, being bigger than 2, and also for the even number 8 in our example, that this even number can be written as the sum of two prime numbers (each of them being bigger than 2).
Remark: we leave the prime number 2 into the play in order to protect the simplicity of our example for the time being.
We now do know that the first one of both intended prime numbers (being p1) which will always equal the number 2, will be part of the 'first half' of the number 8, while the second one, which we will call c1 (c indicating that this is a complement and the index 1 indicating the fact that in here it is about the complement of p1, being the number that must be added to p1 in order to get E as a result of this addition), and which will always equal the number (E-2), will be part of the 'second half' of the number 8. In other terms: concerning p1 it will hold that: : 0<p1<or= 4 and concerning c1, it will hold that: 4 <or= c1 < 8. So, for each number pi it will hold that 0<pi<or= 4 and for each number ci it will hold that 4<or=ci< 8. Concerning the graphic image used in here, this means that pi will always be situated somewhere on the first half of the piece of line which is representing the even number E, while ci will always be situated somewhere on the second half of that piece of line. As a matter of fact, the middle of that piece of line is being concerned as making part of both the first and the second half of that piece of line.
We repeat: the Goldbach Conjecture holds that, concerning each possible E, there must exist a pi that has at least one prime complement ci, which we will express here by qi.
We now restrict things to our example, and so we can write: the Goldbach Conjecture means that the number 8 (as well as whatever even number that is bigger than 2) can be written, either as 2+c1 , or as 2+c2 , wherein either c1 or c2 is a prime number, in other terms: is a qi. (Remark: in these, the numbers 2 and 3 are the prime numbers coming from the first half of the number E=8, and the c1 and c2 are numbers coming from its second half - and at least one of these two numbers has to be a prime number in order to consolidate the Goldbach Conjecture).
We now consider, on our representation of the number 8, the half of 8 (being the number 4) as a 'mirror'. In general, this mirror equals the number (E:2).
In doing so, we can observe c1 (wherein c1=E-2) being the reflection (through the indicated mirror) of the number 2, and c2 (wherein c2=E-3) being the analogue reflection of the number 3. This is so because we do know that the respective sums of pi and ci in both cases must equal E.
On our representation of the number 8, we now indicate these mirror-images by the character "C", as follows:
So, what the Goldbach Conjecture says, is this: "at least one of the C's that are been generated in this way, shall be a prime number again (- and this holds concerning every even number that is bigger than the number 2)."
At this time, we act as if we did not know which numbers qi , so that 4<or=qi<8 , were prime numbers.
We consider again our representation of the number 8, and we indicate on it all numbers laying between 0 and 8 which can never be prime numbers; these are well-defined the compound numbers and, more explicitly: these are the multiples of the prime numbers out of the first half of our fragment; so these are the multiples of the prime numbers pi , so that 0<pi <or= 4, which have already been indicated. Let us remark that only two kinds of numbers exist, being: the prime numbers and the compound numbers. The latter are the multiples of the prime numbers.
We can find these multiples by drawing waves that start each time at the number 0 and that cross each one of these prime numbers as follows:
Remark again that these 'waves' do not indicate physical waves, for they are only used as didactic expedience in order to get a clear representation of numbers simultaneously being composed out of terms and factors.
The waves, originating at 0, that cross a specific P, factually throw all multiples forward as in a whip-lash, and more specifically they do so repeatedly at the crossing-points of each wave at the horizontal axis.
So here we get two waves, namely: (1°) the wave of the prime number 2 (the full line), that indicates the multiples of 2 at each crossing-point at the axis and, (2°) the wave of the prime number 3 (the dotted line), that indicates the multiples of 3 at each crossing-point at the axis.
In this way we can clearly see:
(1°) that the number 4 cannot be prime due to the wave of 2;
(2°) that the number 6 cannot be prime due to the wave of 2;
(3°) that the number 6 cannot be prime due to the wave of 3;
(4°) that the number 8 cannot be prime due to the wave of 2.
Let us repeat:
all those numbers on the right hand side half of our representation of the number 8, that are been crossed by one of our prime number waves coming from the left hand side half of that representation, cannot be prime, because these are multiples of prime numbers. Our representation shows us that this is the case concerning the numbers 4, 6 and 8, for these are compound numbers.
Let us already remark that all numbers left, and belonging to the right hand sided half of the fragment representing our number, will be prime numbers because there exist no third kind of numbers apart from the compound numbers and the prime numbers (and the number 1, of course).
Now we apply the following manner of representation: in order to integrate the mentioned 'process of mirroring' into our 'wave-method', (in our example concerning the number 8) we will not only mirror the prime numbers pi (so that 0<pi <or= 4) throughout 4, but, moreover, we will mirror the waves just generated. When drawing these mirrored waves in red colour, our representation of the number 8 looks as follows:
The waves going from the Left to the Right hand side (here coloured in black) will be called 'LRwaves'. The waves going from the Right to the Left hand side (here coloured in red) will be called 'RLwaves'.
As one can see, the LRwaves depart from 0 and, because their mirror-images are the RLwaves, these RLwaves depart from E, due to the fact that E is the mirror-image of 0.
Again: the RLwaves are generated by the mirroring of the LRwaves throughout (E:2). [In our representation, (E:2) equals 4]. We remark further on that the number 4 is its own mirror-image. We can also see that each number E that has a prime number as its half, fulfils the demand of the Goldbach Conjecture. (For that reason, such a number E=2pi will be excluded as an example in the supposition that follows immediately).
So, here comes a 'reductio ad absurdum' (and it is very important to understand this well):
If the Goldbach Conjecture were untrue, then at least one even number should exist in the representation of which the mirror-images of the prime numbers pi (so that 0<pi <or= 4) would never be prime. In other terms: concerning that number, all mirror-images of the prime numbers pi out of the first half of our number, would be situated on LRwaves..(*)
(*) In other terms: if Goldbach were untrue, it would follow that there would be at least one E that could not be written as a sum of two primes; for now we know that the first term of that sum necessarily originates from the first half of our fragment, while its second term originates from the second half of it; the supposition that such an E could never be written as a sum consisting of two primes, would mean that not one term out of the first half of the fragment representing E would have a mirror-image that would be prime too; but if that second term could never be prime, it would always be compound, and so it would always lay on at least one of the LRwaves.
For the LRwaves always cross the horizontal axis in the second half of the fragment representing our number, at points which are multiples of the prime numbers out of the first half of the fragment representing our number. [Remark that, in this way, we got all compound numbers in the second half of the piece of line that is representing the number E .]
Though it is clear that this can only be the case on the condition that this even number (- being the supposed number E that would contradict the Goldbach Conjecture... if it should exist!) were such that the RLwaves would mirror the LRwaves (because in each case the sum of mirror-images forms the respective even number) - in other terms: if all LRwaves would coincide with the RLwaves. [In order to take away definitely all possible doubts, we will offer a supplementary explanation concerning this matter in paragraph B.2.]
Firstly, let us remark another thing in order to avoid misunderstandings: the bowing of the mentioned waves, either upwards or downwards, is of no importance for, as yet has been said, in here we do not aim physical waves, but just a mere didactic representation; consequently, the waves on the upside of the horizontal axis must be considered as being identical with the waves on the downside of it, as soon as they cross the same points (numbers) on the horizontal axis.
For now, let us suppose that, concerning a considered even number E bigger than 2, all LRwaves would indeed coincide with all RLwaves, then this would mean that the number in question (- and, for now, let us consider the representation of our example of the number 8) had to contain all prime number factors being either smaller than 4 or equal to 4.
In general: supposed that, concerning a given even number E bigger than 2, all LRwaves would indeed coincide with all RLwaves, then this would mean that the number in question had to contain all prime number factors being either smaller than (E:2) or equal to (E:2). For all these LRwaves, if they are being mirrored into RLwaves, will cross the horizontal axis in E; in other terms: they will also 'arrive' in E.
Now we can see the following: to fulfil this condition, concerning a value of E wherein E>8, this E should have to be bigger than E (sic!), for the reason that yet the product of all prime numbers pi , so that 0<pi <or= (E:2), is always bigger than E itself, as can be demonstrated easily by the means of Erdös’s/Tschebycheff's thesis.
Remark: the cases in which E <or= 8 as a matter of fact can be handled separated on their own.
So we must conclude that the Goldbach Conjecture cannot be untrue, which was to be demonstrated.
Let us repeat all this briefly:
The Goldbach Conjecture were untrue if an even number E should exist, so that all ci [being mirror-images over (E:2) of the prime numbers pi , so that 0<pi <or= (E:2)] were multiples (more specifically multiples of pi). For in that case no sum consisting of the terms pi and qi , both prime, could be found. Now, the numbers situated between (E:2) and E are either prime numbers, or multiples of prime numbers - there is no third possibility. We know for sure that all multiples are situated on LRwaves which, as we know, throw the multiples of the prime numbers forwards in a whip-lash into the infinite. So we can express this also by saying: the Goldbach Conjecture were untrue if there should exist an even number E, so that all qi [being the mirror-images of the prime numbers pi so that 0<pi <or= (E:2)] were situated on the LRwaves (more specifically: either equal to (E:2) or between (E:2) and E, which means: waves situated on the line-fragment [(E:2),E]), due to the fact that in that case no one of these mirror-images would be prime and, consequently, no sum of two prime numbers ever could equal E. For now, there would be no problem if the LRwaves, once beyond (E:2), would reflect themselves, in this very sense that their forms either at the right hand side or at the left hand side of (E:2) would be the same, for in this case we would know for sure that all qi would reflect all pi throughout (E:2), and that all these qi would be multiples of prime numbers, because in that very case they should be situated on the LRwaves, either at the right or at the left of (E:2). Though the very problem is this: the forms of the waves either at right or at left of (E:2) are not necessarily each others mirror-images (and as will be demonstrated, they factually never are, but this we do not know at this very moment). Though, in order all pi to be reflected in qi , they yet have to be each other's mirror-images. Well, they could be indeed, namely in the one restricted case in which the image of the forms of the LRwaves situated either between 0 and (E:2), or equal to (E:2) (- these are the waves situated on the line-fragment [0,(E:2]), mirrored in (E:2), would coincide with the image of the forms of the LRwaves in the way they look, once they either coincide with (E:2), or are beyond it (- these are the waves situated on the line-fragment [(E:2),E]). So we should try to imagine the existence of an even number E (on a representation analogue to the representation of our example) wherein the LRwaves coincide perfectly with their reflections throughout (E:2), and these reflections have been called RLwaves. In such a representation, all RLwaves will depart necessarily from E, because E is the reflection of 0, out of which all LRwaves depart. So, if the LRwaves coincide with the RLwaves, this means that all LRwaves (as has been said: departing from 0) will arrive at E. This means that in that very case, E will have to contain all prime number factors pi . Though, in order this case to be possible, the number E (from a value E>8 on) will always be to small: as has been said, this can easily be demonstrated by the means of Erdös/Tchebycheff's thesis, which states that there is at least one prime number between each number and its twofold. In this way we may conclude that the supposition that would make the Goldbach Conjecture untrue, never can be true itself. What was to be demonstrated.
In the right half of our number E, all compound numbers are being marked by LRwaves.
(N.B.: LRwaves originate from the primes out of the first half of our number. We know for sure that these waves mark all of the compound numbers in the second half of E, because the twofold is the smallest composition while we are considering halfs of numbers in here. So it is impossible that in the second half of E, another 'point' would arise apart from eighter a prime or a multiple of a prime out of the first half of E; in other terms: each compound number in the second half of E shall necessarily be a multiple of a prime from the first half of E. And apart from primes and compound numbers, no other numbers in our second half of E do exist).
Now, Goldbach were untrue if at least one well-defined E* would exist, being so that it holds concerning all possible sums pi+ci=E* that ci were compound.
Now it is given that all ci are mirror-images of pi, resulting in the fact that all ci are situated on RLwaves, and we know as well that, concerning each possible E, the number of ci equals the number of pi.
Now, in supposing that Goldbach were untrue, and so if a E* would exists that would contradict Goldbach, and if, in doing so, it would hold concerning all possible sums pi+ci=E* that ci is compound, then it would follow that, in our representation of E*, all ci not only will be situated on RLwaves but simultaneously they will be situated on LRwaves.
Because, concerning each E, the number of pi will equal the number of ci , it will hold that concerning the E* that contradicts Goldbach, its LRwaves necessarily will coincide with its RLwaves.
Yet, referring to Tschebycheff, such an E* cannot exist, and so Goldbach cannot be contradicted.
(to be continued)