As I understand it, the Goldbach conjecture states…
Any number n larger than 2 is the sum of two primes.
This statement was bounced around by Gauss, Euler, amongst others (Presumably Goldbach also, whoever he is)
Since the 4-color map problem and Fermat’s last theorem has been solved, this is apparently the only major math problem left unsolved.
I can say any number is the sum of two numbers.
What’s the big deal about Goldbach?
Well, IANAM (I am not a Mathemetician) but there are lots of easy counterexamples the conjecture you state.
Examples:
3 (No, 1 isn’t prime, nor is zero.)
11,17… (more generally, any odd non-prime plus 2)
Is Goldbach’s conjecture actually limited to even numbers?
From the britannica:
As for the OP – why does a mountain climber climb a mountain?
Define “major”. There are a great many unsolved math problems out there, and more unsolved conjectures being formulated all the time. See, for example,
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html
http://www.math.purdue.edu/~eremenko/uns.html
Has it been mentioned that it hasn’t been proven? O yeh, in the OP. But it is so simply stated. Very enticing.
Another one along the same lines is the Collatz. Hmm. We had a thread a month or two ago about tough math problems, didn’t we?
In the early part of the 20th-century, David Hilbert made a list of mathematical problems that he thought worth working on. They are called Hilbert’s 21 (or some other number) Problems. They are very interesting, but most of them aren’t understandable unless you have a degree in higher mathematics. Which brings me to my hijack: IS IT REALLY IMPOSSIBLE TO STATE MATHEMATICAL PROBLEMS SO THAT PEOPLE WITH AVERAGE INTELLIGENCE CAN UNDERSTAND THEM?
Or is it not, because at some point in the course of learning mathematics, a person learns terms and takes certain things for granted so that he or she cannot even imagine what terms he or she used to think of when he or she wasn’t mathematically informed, and therefore he or she cannot pass on a simple explanation? I finally learned what the Euler phi or totient function was, but it took a long time and yet now that I know it, there is nothing simpler. And I’m sure I could explain it in laymen terms. Thus I am not across that border of a subject beyond which there can be no communication from the knowers to the uninitiated.
It is as if people who know advanced mathematics and physics have died and cannot be contacted, nor can they contact us.
They have joined the Divine in a mystical union guaranteeing the ignorance of the remainder of the population.
The conjecture as stated in the OP isn’t true. For that to be true, as (Tim) stated, all odd numbers would have to be prime (because the only way to sum two numbers and get an odd is if one is even and one is odd, and the only even prime is 2, therefore every odd number is the sum of an odd prime and 2), which, obviously, is not so (ex.: 3x7=21, so 23 cannot be a sum of two primes).
According to this site, Goldbach’s Conjecture is this:
This site (item #98), however, says this:
…While this site says:
…Which are simply several different ways of stating the same thing.
I wish I had half a clue how to even BEGIN that proof… sigh To answer the OP, it’s a big deal simply because it’s “neat” and because it’s unproven. That’s what mathematicians do: Try to solve unsolved problems.
That reminds me, I seem to recall having heard a related conjecture (theorem?) stating that all perfect squares can be written as the sum of two primes. Proven? Not? Misstated? I can’t seem to find it because I can’t recall who it is named after.
I’d say the biggest unsolved math problem right now is the Riemann Hypothesis. There’s a whole structure of results based on the Riemann hypothesis being true, and a whole structure based on it being false; once that’s settled, a lot of theory will fall in to place.
If I’m not mistaken, you must have misstated it.
One corollary of the theorem as you stated it is that if you subtract two from the square of any odd number, you get a prime.
This means that the formula (2n+1)^2 - 2 = p gives prime p for all n.
IIRC ring theory says that you can’t get all primes from a formula as simple as that.
-Ben
Settling Goldbach’s Conjecture would be neat just because the problem’s been so intractable for so long. But neither a proof nor a counter-example would be of any lasting value unless it resulted in new math. That Andrew Wiles’ work linking elliptic curves and modular forms resulted in a proof of Fermat’s Last Theorem is negligible compared to the new techniques he developed and the research avenues he iluminated. That’s his great accomplishment.
Cabbage said:
No argument. Maybe that’s what HoldenCaulfield has resolved.
Yeah I figured out what it was I was thinking of. Every odd number can be expressed as the difference of two consecutive squares:
2n+1=(n+1)[sup]2[/sup]-n[sup]2[/sup]
That one is trivial to prove, so I’m kind of disappointed. I had remembered it as being more intriguing than that.
don willard:
I think it’s not impossible, but it does take a rare person who is able to understand that stuff and be able to explain it to the rest of us.
Math is not religion, no matter how much people say it is not a science.
I’m pretty sure that most (all?) of the “people who know advanced mathematics” could explain the same in laymen’s terms, as you can. Your own experience should convince you that it is just a continuum. Stuff gets complicated.
Depending on what you mean by “average”, of course it isn’t impossible. Authors like Hofstadter and Dunham seem to do a fairly good job in making some of the great ideas of mathematics accessible to the layman. But that’s not to say these are easy books for the non-mathemetician. They take considerable perseverance, and the reader, even if not educated in higher math, is likely to be above average in intelligence than the average Joe. What I’ve concluded from these books is that there are many fascinating principles in the field that can be taught to someone without a lot of prior preparation. But that doesn’t mean you can teach Calculus to someone who hasn’t learned Algebra.
don:
The concepts aren’t hard, but the type of thinking is. Most people don’t normally think in the step-by-step logical way needed to work up a rigorous proof. But the concepts are all founded, ultimately, upon the basic concepts of arithmetic: 1+1=2, 2+1=3, 2*2=4, etc. All of them based on observation. (I have one apple. I steal one apple from the local dolt. I now have two apples. Therefore, one plus one is two.) Most people never take them to the logical conclusions of calculus or non-Euclidian geometry, but you can.