When I first heard of Goldbach's conjecture in maybe 2003, I had recently been studying image compression. And I immediately saw it as an information theory-type problem. So the proof would be a proof that you could not describe the position of primes on a number line relative to the left end, with multiples of (smaller) prime distances from the right. You would miss some, like a compressed image with lost quality. But I didn't know how to prove this.But thinking about this for a second, it seems so obvious as to be considered proven, if only in the way things are proven every day by non-mathematicians.What you are describing is a set of distances that are not multiples of each other, or which are unique multiples of a single distance. It doesn't matter if you vary the directions, on a one-dimensional number line, in three dimensions, or just consider them as relative values absent of space, or what. Put simply, you cannot describe a set of distances with a smaller set of distances. And most prime distances going one way, specifically those larger than half the number, cannot be used as multiples going the other way, as they will not become multiples until outside of the relevant interval.If you consider the even number 30, you cannot use distances 3,5,7,11,13 to describe the positions of 3,5,7,11,13,17,19,23,29 from a single point going the opposite direction. Is this not obvious?You might try to think of how, well, maybe from 17 to 23 could be like a ratio of 5/3 to 13/11. But it is obvious that as you get further from zero, the precision increases, meaning each increment of 1 is a smaller percent change in the ratio. You cannot use the larger percentage changes available in the smaller ratios, to describe the fine percentage changes which by definition fall between combinations of them.Again, I have never written a mathematician's formal proof in my life, but to me Goldbach's conjecture is obvious.

"Obvious is the most dangerous word in mathematics."

QuoteOriginally posted by: Ruairi"Obvious is the most dangerous word in mathematics."I am not a mathematician, but "obvious" saves me from driving head-on into walls a thousand times a day.I don't now what math is, it seems like a deep philosophical question. But at a quick glance, it seems that "proving" things might simply amount to extending the set of rules, by describing new rules, or whatever.It may be possible that mathematics cannot prove Goldbach's conjecture. Because to prove anything, such as that I will have to turn left to avoid a wall, requires plugging in facts, and applying mathematics to them. And perhaps there is simply no way to plug in obvious facts about the ratios of infinite unique prime numbers into a single equation.So maybe it is not a problem of proving it, but expressing the proof in a math way. If you had to prove in court mathematically that somebody was guilty of murder, I am not sure you could do it for every case in which someone is convicted beyond a reasonable doubt.So while the reason why every even number is the sum of two primes may be obvious - as obvious as who the killer is when the victim's sister and the neighbor saw him - there may still exist a problem with plugging the facts into some kind of math equation that can accommodate them.

Never mind all that, because there is a possibility someone is not the murderer. But to me, this is as obvious as 7 not being equal to 5. Can you prove the you can't drive seven cars with only five people? It is obvious you cannot drive seven cars with five people. But you have to accept the assumption that it takes one person to drive a car. If this is not part of the list of mathematical rules, then you can't prove it.

n > 0X+n != XX distances < X+ n distancesYou cannot tell someone 7 distances, in a letter that only mentions 5 distances. It does not matter how you describe or record them, onto what they are projected, or in what manner they manifest.If they are distances on a line, such that there are rules of transitive distance between two points and a third point, it makes no difference. If they are the complement of a collection of distances, left over when you subtract that set of distances from a single distance, it makes no difference.When you talk about a sum of two primes, and those two primes represent two distances, meaning a distance and its complement, you cannot get a larger set of complements, out of a smaller set of distances and their permutations, where those distances have been pre-selected to not be multiples or permutations of each other.

You can turn it around, and say prime numbers are the numbers which fit Goldbach's conjecture. They are the set of numbers that represent distances which cannot be recombined to get any other member of the set, or to get a greater set. So that when they are used alone or in combination to represent one distance from a point, there is no number of hops of that number that can get you to another. If you have a smaller set of these Goldbachian numbers, you lack the precision to hop to all the points in the larger set of distances to a single point.

QuoteOriginally posted by: farmerYou can turn it around, and say prime numbers are the numbers which fit Goldbach's conjecture. They are the set of numbers that represent distances which cannot be recombined to get any other member of the set, or to get a greater set.Any numbers they can be recombined to get, are excluded from the set. Any numbers beyond the minimum set required to express the distance from two origins in the opposite direction, shall be labeled "unprime."The prime numbers are those whose sums and combinations are required to express all distances to equidistant points between two origins. If there are redundant or extraneous distances, that can be constructed out of the smaller set of distances, we discard them. Any point distance that can be described as a combination of smaller distances, is removed from the set. So we keep removing numbers from our set UNTIL the even number is a sum of two members of this set which are in use and cannot be discarded, and we will then call this residual set "prime." So composing minimum sets of numbers to describe distances to equidistant points - some of which distances are combinations of rougher ratios - is an exercise to arrive at the prime set.If an even number is not the sum of two primes, you reclassify each number as prime or not prime until it is. So Goldbach's conjecture is simply a convoluted way to define the set of prime numbers less than a given even number.

Put even more simply, we can say a prime number is a number in non-redundant use to describe a distance somewhere in the number line.So Goldbach's conjecture is merely a word trick. It is saying that every set of distances can be described with a minimum set of nonredundant numbers, and when we have found this set for a given set of distances, we have found the set which we need and cannot use something less than.Every even number is the sum of two numbers which are in the minimum set we need to add up to the even number. Obviously...

- Traden4Alpha
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Couldn't we say the same about other number sequences? For example, we might postulate that any whole number > 2 is the sum of two Fibonacci numbers. It certainly seems true for low N but fails for N=12. I'm sure one can find other number sets that seem to provide pair-wise sums to create all even numbers, odd numbers, or whatever numbers but that fail at high N.The issue with Goldbach's conjecture is the potential that somewhere in the high primes lies some sparse desert of values or coincidental pattern in the numbers that prevents finding a pair of primes that some to a particular even number.The prime set is the minimum set for constructing all whole numbers using only the multiplication operator. That this set has other properties under addition does not seem obvious.

QuoteOriginally posted by: Traden4AlphaCouldn't we say the same about other number sequences? For example, we might postulate that any whole number > 2 is the sum of two Fibonacci numbers.No, you have missed the point completely. Fibonacci numbers have nothing to do with the possible sums, or all combinations, of two numbers that add up to a third number, and the relative distances from the origin. Prime numbers are specifically the minimum non-redundant set required to describe a set of distances, which is the same set of distances that must be described in two directions by a set of numbers when you are talking about all sums of two numbers that make a number.I don't even know what fibonnaci numbers are, or what they describe. But they have not been selected for their ability to describe all points on the number line non-redundantly. Really Goldbach's conjecture is just another way of defining prime numbers. It is a finite way of defining prime numbers less than an even number, whereas the algorithm for arriving at all primes has no finite definition.When you go up the number line, and you need to add a new number to describe the next point because you cannot describe it with combinations of previous numbers, you call that number a prime. This is just a rearrangement of the problem of describing every whole number distance between to points at an even distance in two directions. It is the same problem, and prime numbers, not fibonnaci numbers, are the solution to this problem. Fibonnaci numbers are the solution to some other problem, I don't know anything about it.

QuoteOriginally posted by: farmer...Prime numbers are specifically the minimum non-redundant set required to describe a set of distances, which is the same set of distances that must be described in two directions by a set of numbers when you are talking about all sums of two numbers that make a number....I thought powers were supposed to do that, not prime numbers ... what do you mean by the phrase above ?

QuoteOriginally posted by: MCarreiraI thought powers were supposed to do that, not prime numbers ... what do you mean by the phrase above ?Primes are the minimum set whose powers can describe all the necessary distances.

QuoteOriginally posted by: farmerQuoteOriginally posted by: MCarreiraI thought powers were supposed to do that, not prime numbers ... what do you mean by the phrase above ?Primes are the minimum set whose powers can describe all the necessary distances.But the Goldbach conjecture does not use powers of primes ... so I'm not sure that the property above will be enough to prove it (or be used to provide the intuition to make sure that a certain property will always hold).

QuoteOriginally posted by: MCarreiraBut the Goldbach conjecture does not use powers of primesSure it does. It uses primes, and products of primes, some of which are powers of primes, and all of which are products of powers of primes, to describe every whole number distance between two evenly-spaced numbers, from two directions. To do so, it must use at least one prime to describe every spot described by a prime going in the other direction.Most importantly, it uses products of powers of primes to determine which numbers are prime and which are not! You need to use 3x3, for example, to determine whether 9 is prime. You need to go through all your products of powers to determine if 11 is prime. Prime can be described as not a product of powers of other primes. Prime is simply what you need when you run out of products of powers.

QuoteOriginally posted by: farmerQuoteOriginally posted by: MCarreiraBut the Goldbach conjecture does not use powers of primesSure it does. It uses primes, and products of primes, some of which are powers of primes, and all of which are products of powers of primes, to describe every whole number distance between two evenly-spaced numbers, from two directions. To do so, it must use at least one prime to describe every spot described by a prime going in the other direction.Most importantly, it uses products of powers of primes to determine which numbers are prime and which are not! You need to use 3x3, for example, to determine whether 9 is prime. You need to go through all your products of powers to determine if 11 is prime. Prime can be described as not a product of powers of other primes. Prime is simply what you need when you run out of products of powers.You probably mean this.

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