Goldbach Conjecture: Closer To Solved? 170
mikejuk writes "The Goldbach conjecture is not the sort of thing that relates to practical applications, but they used to say the same thing about electricity. The Goldbach conjecture is reasonably well known: every integer can be expressed as the sum of two primes. Very easy to state, but it seems very difficult to prove. Terence Tao, a Fields medalist, has published a paper that proves that every odd number greater than 1 is the sum of at most five primes. This may not sound like much of an advance, but notice that there is no stipulation for the integer to be greater than some bound. This is a complete proof of a slightly lesser conjecture, and might point the way to getting the number of primes needed down from at most five to at most 2. Notice that no computers were involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search."
Re:and here is the proof for every even number (Score:5, Insightful)
I hereby prove that every even number is a sum of no more than six primes, one of those is 1.
Psst, 1 isn't prime. Or composite. It's neither.
True, but you can change the GP's proof to "every even number n (where n > 4) is a sum of no more than six primes, because m = n - 3 is an odd number".
Re:Every Integer? (Score:5, Insightful)
7 + 2 = 9
The numbers less than 3 (Score:4, Insightful)
If you're talking about integers (which this conjecture refers to), then that's easy:
2 = 5 + -3
0 is trivial:
0 = p + -p for all prime numbers p
1 is also fairly easy:
1 = 3 + -2
And just to complete this, here's 3:
3 = 5 + -2
[multiplication by a unit, in this case -1, does not change the "primeness" of a number]
Is this progress? (Score:5, Insightful)
Sorry, but I can't accept this being progress toward a proof.
Consider Fermat's Last Theorem. Proving it for any particular exponent is doable. Mathematicians had proved it for various sets of exponents (Sophie Germain, Wieferich, etc.). But the proof for all exponents was based on completely different mathematics (Elliptic curves/modular forms, Taniyama-Shimura, Wiles) and didn't look like anything that had come before.
...laura
Re:It's every *even* number (Score:4, Insightful)
Actually "infinity" is an honest number in several modern, rigorous senses.
In the extended real numbers [wikipedia.org], one adds two symbols to the usual real numbers (which won't render here), "+inf" and "-inf". No mystical qualities are needed; one could just as well use symbols "@" and "#". The extended real numbers are useful in formulating elementary measure theory, where some basic arithmetic with them is defined (+inf - -inf = +inf, for instance; +inf + -inf is left undefined).
In the real projective line [wikipedia.org], one adds a single "point at infinity" which is imagined to "wrap around" from "negative infinity" to "positive infinity". I'm sorry for all the scare quotes; the actual construction is rigorous. Suppose you have a plane and a horizontal line passing through y=1. Given a point on the horizontal line, there is another line passing through that point and the origin; this line is taken to be a "point" on the real projective line. The additional point at infinity is taken to be the horizontal line passing through the origin, which is the limiting value of the other real projective line-points as they go to positive or negative infinity.
As for X/X = 1 vs. X/0 = infinity when X=0, one could simply say X/0 = infinity when X is not 0 and then there is no conflict. But again, the usual rules of arithmetic don't work well in this situation, so you need a good reason to extend arithmetic to work with infinities. The only case I've encountered where that is true is with the extended real numbers in measure theory mentioned above.
As for mathematicians, yes, we change conventions whenever needed without real difficulty. The phrase "ring" is a great example--it can have a huge variety of meanings depending on context. Careful authors will specify, but otherwise you'll have to figure out from context what precisely is meant. Once in a while this can be confusing, but for something as simple as whether primes can be negative or not it's a complete non-issue.