Gödel, Escher, Bach: An Eternal Golden Braid (70 page)

Achilles: What a strange result. Of what good is it?

Tortoise: It has brought the problem into the domain of the finite. Previous to Schnirelmann's proof, it was conceivable that as you took larger and larger even numbers, they would require more and more primes to represent them. Some even number might take a trillion primes to represent it! Now it is known that that is not so-a sum of 300,000 primes (or fewer) will always suffice.

Achilles: I see.

Tortoise: Then in 1937, a sly fellow named Vinogradov-a Russian too-managed to establish something far closer to the desired result: namely, every sufficiently large ODD number can be represented as a sum of no more than THREE odd primes. For example, 1937 = 641 + 643 + 653. We could say that an odd number which is representable as a sum of three odd primes has "the Vinogradov property. Thus, all sufficiently large odd numbers have the Vinigradov properties

Achilles: Very well-but what does "sufficiently large" mean?

Tortoise: It means that some finite number of odd numbers may fail to have the Vinogradov property, but there is a number-call it 'v'beyond which all odd numbers have the Vinogradov property. But Vinogradov was unable to say how big v is. So in a way, v is like g, the finite but unknown number of Goldberg Variations. Merely knowing that v is finite isn't the same as knowing how big v is. Consequently, this information won't tell us when the last odd number which needs more than three primes to represent it has been located.

Achilles: I see. And so any sufficiently large even number 2N can be represented as a sum of FOUR primes, by first representing 2N - 3 as a sum of three primes, and then adding back the prime number 3.

Tortoise: Precisely. Another close approach is contained in the Theorem which says, "All even numbers can be represented as a sum of one prime and one number which is a product of at most two primes."

Achilles: This question about sums of two primes certainly leads you into strange territory. I wonder where you would be led if you looked at DIFFERENCES of two odd primes. I'll bet I could glean some insight into this teaser by making a little table of even numbers, and their representations as differences of two odd primes, just as I did for sums. Let's see ...

2= 5-3,

7-5,

13-11,

19-17, etc.

4 = 7 - 3,

11 - 7,

17 - 13,

23 - 19,etc.

6 = 11 - 5, 13 - 7,

17 - 11,

19- 13, etc.

8 = 11 - 3, 13 - 5,

19 - 11,

31 - 23,etc.

10 = 13 - 3, 17 - 7,

23 - 13,

29- 19, etc.

My gracious! There seems to be no end to the number of different representations I can find for these even numbers. Yet I don't discern any simple regularity in the table so far.

Tortoise: Perhaps there is no regularity to be discerned.

Achilles: Oh, you and your constant rumblings about chaos! I'll hear none of that, thank you.

Tortoise: Do you suppose that EVERY even number can be represented somehow as the difference of two odd primes?

Achilles: The answer certainly would appear to be yes, from my table. But then again, I suppose it could also be no. That doesn't really get us very far, does it?

Tortoise: With all due respect, I would say there are deeper insights to be had on the matter.

Achilles: Curious how similar this problem is to Goldbach's original one. Perhaps it should be called a "Goldbach Variation".

Tortoise: Indeed. But you know, there is a rather striking difference between the Goldbach Conjecture, and this Goldbach Variation, which I would like to tell you about. Let us say that an even number 2N has the “Goldbach property” if it is the SUM of two odd primes, and it has the “Tortoise property” if it is the DIFFERENCE

of two odd primes

Achilles: I think you should call it the "Achilles property". After all, I suggested the problem.

Tortoise: I was just about to propose that we should say that a number which LACKS the Tortoise property has the "Achilles property". Achilles: Well, all right .. .

Tortoise: Now consider, for instance, whether I trillion has the Goldbach property or the Tortoise property. Of course, it may have both.

Achilles: I can consider it, but I doubt whether I can give you an answer to either question.

Tortoise: Don't give up so soon. Suppose I asked you to answer one or the other question.

Which one would you pick to work on?

Achilles: I suppose I would flip a coin. I don't see much difference between them.

Tortoise: Aha: But there's a world of difference' If you pick the Goldbach property, involving SUMS of primes, then you are limited to using primes which are bounded between 2 and 1 trillion, right?

Achilles: Of course.

Tortoise: So your search for a representation for 1 trillion as a sum of two primes is GUARANTEED TO TERMINATE.

Achilles: Ahhh! I see your point. Whereas if I chose to work on representing 1 trillion as the DIFFERENCE of two primes, I would not have any bound on the size of the primes involved. They might be so big that it would take me a trillion years to find them.

Tortoise: Or then again, they might not even EXIST! After all, that's what the question was asking-do such primes exist, It wasn't of much concern how big they might turn out to be.

Achilles: You're right. If they didn't exist, then a search process would lead on forever, never answering yes, and never answering no. And nevertheless, the answer would be no.

Tortoise: So if you have some number, and you wish to test whether it has the Goldbach property or the Tortoise property, the difference between the two tests will be this: in the former, the search involved is GUARANTEED TO TERMINATE; in the latter, it is POTENTIALLY ENDLESS-there are no guarantees of any type. It might just go merrily on forever, without yielding an answer. And yet, on the other hand, in some cases, it might stop on the first step.

Achilles: I see there is a rather vast difference between the Goldbach and Tortoise properties.

Tortoise: Yes, the two similar problems concern these vastly different properties. The Goldbach Conjecture is to the effect that all even numbers have the Goldbach property; the Goldbach Variation suggests that all even numbers have the Tortoise property. Both problems are unsolved, but what is interesting is that although they sound very much alike, they involve properties of whole numbers which are quite different.

Achilles: I see what you mean. The Goldbach property is a detectable, or recognizable property of any even number, since I know how to test for its presence just embark on a search. It will automatically come to an end with a yes or no answer. The Tortoise property, however, is more elusive, since a brute force search just may never give an answer.

Tortoise: Well, there may be cleverer ways of searching in the case of the Tortoise property, and maybe following one of them would always come to an end, and yield an answer.

Achilles: Couldn't the search only end if the answer were yes%

Tortoise: Not necessarily. There might be some way of proving that whenever the search lasts longer than a certain length of time, then the answer must be no. There might even be some OTHER way of searching for the primes, not such a brute force way, which is guaranteed to find them if they exist, and to tell if they don't. In either case, a finite search would be able to yield the answer no. But I don't know if such a thing can be proven or not. Searching through infinite spaces is always a tricky matter, you know.

Achilles: So as things stand now, you know of no test for the Tortoise property which is guaranteed to terminate-and yet there MIGHT exist such a search.

Tortoise: Right. I suppose one could embark on a search for such a search, but I can give no guarantee that that "meta-search" would terminate, either.

Achilles: You know, it strikes me as quite peculiar that if some even number-for example, a trillion-failed to have the Tortoise property, it would be caused by an infinite number of separate pieces of information. It's funny to think of wrapping all that information up into one bundle, and calling it, as you so gallantly suggested,

"the Achilles property" of 1 trillion. It is really a property of the number system as a

"HOLE, not just of the number 1 trillion.

Tortoise: That is an interesting observation, Achilles, but I maintain that it makes a good deal of sense to attach this fact to the number 1 trillion nevertheless. For purposes of illustration, let me suggest that you . consider the simpler statement "29 is prime".

Now in fact, this statement really means that 2 times 2 is not 29, and 5 times 6 is not 29, and so forth, doesn't it?

Achilles: It must, I suppose.

Tortoise: But you are perfectly happy to collect all such facts together, and attach them in a bundle to the number 29, saying merely, "29 is prime"

Achilles: Yes ...

Tortoise: And the number of facts involved is actually infinite, isn't it:, After all, such facts as "4444 times 3333 is not 29" are all part of it, aren't they%

Achilles: Strictly speaking, I suppose so. But you and I both know that you can't produce 29 by multiplying two numbers which are both bigger than 29. So in reality, saying

"29 is prime" is only summarizing a FINITE number of facts about multiplication Tortoise: You can put it that way if you want, but think of this: the fact that two numbers which are bigger than 29 can't have a product equal to 29 involves the entire structure of the number system. In that sense, that fact in itself is a summary of an infinite number of facts. You can't get away from the fact, Achilles, that when you say "29 is prime'-', you are actually stating an infinite number of things.

Achilles: Maybe so, but it feels like just one fact to me.

Tortoise: That's because an infinitude of facts are contained in your prior knowledge-they are embedded implicitly in the way you visualize things. You don't see an explicit infinity because it is captured implicitly inside the images you manipulate.

Achilles: I guess that you're right. It still seems odd to lump a property of the entire number system into a unit, and label the unit "primeness of 29"

Tortoise: Perhaps it seems odd, but it is also quite a convenient way to look at things.

Now let us come back to your hypothetical idea. If, as you suggested, the number 1

trillion has the Achilles property, then no matter what prime you add to it, you do not get another prime. Such a state of affairs would be caused by an infinite number of separate mathematical "events". Now do all these "events" necessarily spring from the same source? Do they have to have a common cause? Because if they don't, then some sort of "infinite coincidence" has created the fact, rather than an underlying regularity.

Achilles: An "infinite coincidence"? Among the natural numbers, NoTHING is coincidental-nothing happens without there being some underlying pattern. Take 7, instead of a trillion. I can deal with it more easily, because it is smaller. 7 has the Achilles property.

Tortoise: You're sure?

Achilles: Yes. Here's why. If you add 2 to it, you get 9, which isn't prime. And if you add any other prime to 7, you are adding two odd numbers, resulting in an even number-thus you again fail to get a prime. So here the "Achilleanity" of 7, to coin a term, is a consequence of just Two reasons: a far cry from any "infinite coincidence". Which just goes to support my assertion: that it never takes an infinite number of reasons to account for some arithmetical truth. If there WERE some arithmetical fact which were caused by an infinite collection of unrelated coincidences, then you could never give a finite proof for that truth. And that is ridiculous.

Tortoise: That is a reasonable opinion, and you are in good company in making it.

However

Achilles: Are there actually those who disagree with this view? Such people would have to believe that there are "infinite coincidences", that there is chaos in the midst of the most perfect, harmonious, and beautiful of all creations: the system of natural numbers.

Tortoise: Perhaps they do; but have you ever considered that such chaos might be an integral part of the beauty and harmony?

FIGURE 71. Order and Chaos, by M. C. Escher (lithograph, 1950).

Achilles: Chaos, part of perfection? Order and chaos make a pleasing unity? Heresy!

Tortoise: Your favorite artist, M. C. Escher, has been known to suggest such a heretical point of view in one of his pictures ... And while we're on the subject of chaos, I believe that you might be interested in hearing about two different categories of search, both of which are guaranteed to terminate.