As advised elsewhere CK is interested in reality. Confirmation Theory indicates that we learn more from an exception than a common confirmation. For example, if I understand all ravens to be black, and have 1000 confirmed sightings of black ravens, then confirmed sighting 1001 of another black raven really doesn’t do much for me. On the other hand a confirmed sighting of a white raven is a big deal. Potentially it could cause me to dramatically revise my understanding of important aspects of reality.

So CK pays a lot of attention to anomalies.

Yesterday I talked about the Human Reaction anomaly, wherein when we react to something we move faster than when we initiate the activity.

Today I’m looking at a mathematical anomaly.

Christian Goldbach March 18, 1690 – November 20, 1764 was a noted German mathematician most remembered today for Goldbach’s Conjecture; one of the oldest, best-known and still unsolved problems in all number-theory and indeed Mathematics as a whole. It is ‘unsolved’ in the sense that no mathematician has been able to show a mathematical proof for the conjecture. That’s over 250 years that a statement that is probably true has remained unproven. Given how many mathematicians and AI mathematical ‘proving’ algorithms are now working in the field of mathematics, that’s really surprising. But many things about mathematics continue to surprise, which is one reason why CK pays attention to the subject.

However, this is not the subject of today’s blog. Goldberg made more than one conjecture about numbers. One of them he stated in a letter  to Euler on 18th November 1752.

In this letter he conjectured that every positive odd integer is either prime or the sum of a prime and twice a square.

Examples:

1. 5 is prime

2. 7 is prime

3. 9 = 1 + (2 times (2 squared))

4. 11 is prime.

5. 13 is prime

6. 15 = 7 + (2 times (2 squared))

7. 17 is prime

8. 19 is prime

9. 21 = 3 + (2 times (3 squared))

10. 23 is prime

looking good so far.

As with his most famous conjecture, Goldbach advised that he believed this conjecture to be true, but hadn’t been able to prove it. Let’s call this Goldbach’s Conjecture (2) to conveniently distinguish it from his most famous conjecture, Goldbach’s Conjecture (1).

Euler was as good as any mathematician who has ever lived. He considered Goldbach’s Conjecture (2) and couldn’t refute. Neither could he prove it to be true.

As with my observations of black ravens, the issue of proof and confirmation arises. How many confirmed sightings of black ravens do I need before I am prepared to believe that the conjecture that:

All ravens are black

is true?

It’s the fundamental question of evidence, confirmation and proof, and it’s a big topic in epistemology and science for good reasons. Unlike the objectionable world of life, mathematics, however, involves closed axiomatic systems. One doesn’t have to rely on messy things like confirming evidences and observations in mathematics. You can logically prove things within axiomatic systems. The only things you can’t prove in a closed axiomatic system are the axioms themselves. Mathematicians do not consider any number of confirming instances to constitute proof. And they are right.

Above we have ten confirming instances for this Goldbach’s Conjecture (2).

Let’s try some more:

11. 97 is prime

12. 99 = 1 + (2 times (7 squared))

OK - 1 isn’t considered to be prime nowadays, because they slightly amended the definition of ‘prime number’. At the time Goldbach was writing 1 was still considered ‘prime’ and therefore this would be a valid confirmation of his conjecture. Now the Goldbach’s Conjecture (2) would have to specifically state ‘prime number or 1’.

13. 711 = 199 + (2 times (16 squared))

Yes- 711 isn’t prime but 199 is.

We can keep doing this all night. In fact if you care to take the all the odd numbers between 1 and 5001 I promise you that you’ll get 2500 confirming instances of this Goldbach’s Conjecture (2). Long before you’ve gotten past the first 1000 I’m sure you’ll be thoroughly convinced that the Goldbach Conjecture (2) is right on the money. Surely mathematics is consistent. If something works mathematically it always works. If it doesn’t work it doesn’t work. Mathematics doesn’t work in generalisations. You don’t have mathematical rules with just one exception do you? Well … OK, you do tend to get exceptions for zero, and one, and infinity, but not otherwise. I mean you don’t find a rule that works for every number except 5993 or something crazy, do you? Do you?

It’s an interesting point, and a very interesting question. However, getting back to Goldbach’s Conjecture (2) he may be right, but it’s still unproven. And those crazy mathematicians won’t accept something as being absolutely true until it is proven. So how can you prove it?

Well Goldberg’s Conjecture (1) is probably right and still unproven after 250 years. So you probably won’t be surprised to hear that after a mere hundred years nobody had managed to prove Goldbach’s Conjecture (2) either. You can come up with thousands and millions and billions of confirming examples that it’s right, but that counts for very little with mathematicians.

Then, after a hundred years where no one had been able to prove Goldbach’s Conjectures (1) or (2) in 1856 Moritz Abraham Stern did something far more interesting. He found an anomaly.

He found that the number 5777 cannot be expressed as the sum of a prime number and twice the square of another number. After 104 years Goldbach Conjecture (2) was busted. It can’t be proven because it isn’t true.

This has to be a joke doesn’t it?. Every odd positive number from 1 to 5775 is either prime or can be expressed as the sum of a prime number and twice the square of another number. But 5777 can’t?

That is absolutely true.

5777 is an anomaly. A rule that holds good for pretty much any odd positive number you can possibly think of if you spend the rest of life thinking of them doesn’t hold true for 5777.

Why not? What’s so special about 5777?

Nothing. Except that it is an exception to Goldbach’s Conjecture (2).

Is it the only exception?

No. If you find one then presumably you can find others.

Really? Like what? How many others have been found after 250 years?

Er. Well (embarrassed) there is just one other we know about. I erm … slipped it to you a little earlier in this blog.

5993 is the only other example of a number that’s a known exception to Goldbach’s Conjecture (2).

So maybe those mathematicians aren’t so crazy when they refuse to believe anything no matter how many confirming instances you can cite. If it hasn’t been proven, you don’t know if it is true.

There might be some ugly little numbers like 5777 and 5993 just sitting there as strange little anomalies that don’t cooperate the way every other number seems to do. Maybe there’s one ugly little number somewhere that doesn’t comply with Goldbach’s Conjecture (1) either. After 250 years no one has found it yet, but maybe ….

And what wisdoms can we learn from this?

I’ll let you think about that for yourself.