But it seems misleading to me. The title implies that an odd number could be the sum of two, three, four or five primes, but two and four are excluded because they will generate an even number, so isn’t it really saying that it can be expressed as the sum of either three or five primes? Anyway, nice proof.
[Via Geek Press]
[Update a while later]
D’oh! As the commenter notes, I’d forgotten that two is prime, and unique in it being an even prime.
two is prime, so for primes p,q, and r all odd we have
2 + p which is odd.
and 2 + (p +q ) + r
since p + q is even, and 2 is even then the resulting number must be odd.
Also note the Goldbach conjecture says every even number is the sum of two primes; while that hasn’t been proven, it has been demonstrated for numbers up to 3.5 x 10^17 (or thereabouts). If the Goldbach conjecture holds, every odd number is actually a sum of just 3 primes (adding the prime 3 to the even number 3 less).
Two and four are still excluded: even+odd+odd=even.
I’m wrong. Noticed just as I hit submit. even+odd+odd+odd=odd.
There is an old joke about a mathematician, a physicist, and an engineer considering the hypothesis that all odd numbers greater than 1 are prime.
Mathematician:
“Hmm, 3 is prime, 5 is prime, 7 is prime, 9… hmm, well 9 is a special case, 11 is prime, 13 is prime… sure, sounds like it works!”
Physicist:
“Hmm, 3 is prime, 5 is prime, 7 is prime, 9… hmm, well 9 is an experimental error, 11 is prime, 13 is prime… sure, sounds like it works!”
Engineer:
“Hmm, 3 is prime, 5 is prime, 7 is prime, 9… 9 is prime, 11 is prime, 13 is prime… sure, sounds like it works!”
For some reason the thread here reminds me of this….
Computer Scientist:
“Hmm, 3 is prime, 3 is prime, 3 is prime, 3 is prime, …
Social Scientist:
“Of course I know what a prime is! And these odd numbers are a danger to social justice! We need to redistribute the prime numbers to ensure everyone has an equal share of the product!”
I don’t quite get some of the comments here.
The title is not misleading. The theorem is asserting that for every odd number, N, there exists a set of five or fewer primes that add up to it.
There exist many many sets of five or fewer primes that *don’t* add up to ay given N. And some of those sets many add up to even numbers. But that’s not important. They don’t add up to N and that’s all that matters. Try another set. The theorem assert that you can always find at least one.
All odd numbers which are primes are trivially solvable. There is a winning set containing just the number itself.
So you’re looking at the the odd numbers which aren’t prime: 9, 15, 21, 25, 27, 33, 35, 39, 45, 49 and so on.
You can’t add up any combination of 3, 5 and 7 to get 9, so pretty obviously 2 is needed as one of your allowable primes in the set (i.e. 2+7).
The only gaps of only 2 in the above list are at 27 and 35. So all the rest of those numbers can be formed by adding 2 to a prime.
How can we get the others?
IF we can assume the the Goldbach conjecture then we can just always add up 3 and two other primes. i.e. 27 = 3 + 24 = 3 + 5 + 19 (or 3 + 7 + 17). 35 = 3 + 32 = 3 + 5 + 27 or 3 + 13 + 19.
But we CAN’T assume the Goldbach conjecture. The authors know about both the even Goldbach conjecture (all even numbers are the sum of at most two primes) and the odd Goldbach conjecture (all odd numbers are the sum of at most three primes).
Those conjectures are thought to be true, but they haven’t been PROVEN, which is the point here.
What has previously been PROVEN is that all even numbers are the sum of at most six primes. That’s why PROVING that all odd numbers are the sum of at most five primes is an advance.
So, what is an example of an odd number that actually needs four or five primes added together to make it, and three won’t do? Can I extend the above list a bit and find one?
No.
Every odd number that has EVER ACTUALLY BEEN TESTED has been found to be the sum of at most three primes. Finding a counter-example would disprove the odd Goldbach conjecture, and that hasn’t happened (yet).
As far as we know, both the even and the odd Goldbach conjecture are true, but they haven’t been PROVEN to be true. What has now been proven is that six primes is enough for even numbers, and five primes is enough for odd numbers.
No doubt people are, as we speak, looking for a PROOF that four primes are enough for any even number.