One of the oldest unsolved problems in mathematics is also among the easiest to grasp. The weak Goldbach conjecture says that you can break up any odd number into the sum of, at most, three prime numbers (numbers that cannot be evenly divided by any other number except themselves or 1). For example:
35 = 19 + 13 + 3
or
77 = 53 + 13 + 11
Mathematician Terence Tao of the University of California, Los Angeles, has now inched toward a proof. He has shown that one can write odd numbers as sums of, at most, five primes—and he is hopeful about getting that down to three. Besides the sheer thrill of cracking a nut that has eluded some of the best minds in mathematics for nearly three centuries, Tao says, reaching that coveted goal might lead mathematicians to ideas useful in real life—for example, for encrypting sensitive data.
The weak Goldbach conjecture was proposed by 18th-century mathematician Christian Goldbach. It is the sibling of a statement concerning even numbers, named the strong Goldbach conjecture but actually made by his colleague, mathematician Leonhard Euler. The strong version says that every even number larger than 2 is the sum of two primes. As its name implies, the weak version would follow if the strong were true: to write an odd number as a sum of three primes, it would be sufficient to subtract 3 from it and apply the strong version to the resulting even number.
Mathematicians have checked the validity of both statements by computer for all numbers up to 19 digits, and they have never found an exception. Moreover, the larger the number, the more ways exist to split it into a sum of two other numbers—let alone three. So the odds of the statements being true become better for larger numbers. In fact, mathematicians have demonstrated that if any exceptions to the strong conjecture exist, they should become increasingly sparse as the number edges toward infinity. In the weak case, a classic theorem from the 1930s says that there are, at most, a finite number of exceptions to the conjecture. In other words, the weak Goldbach conjecture is true for “sufficiently large” numbers. Tao combined the computer-based results valid for small-enough numbers with the result that applies to large-enough numbers. By improving earlier calculations with “lots of little tweaks,” he says, he showed that he could bring the two ranges of validity to overlap—as long as he could use five primes.
Next, Tao hopes to extend his approach and show that three primes suffice in all cases. But that is not likely to help with the strong conjecture. The weak conjecture is incomparably easier, Tao says, because by splitting a number into a sum of three, “there are many, many more chances for you to get lucky and have all the numbers be prime.” Thus, a quarter of a millennium after Goldbach’s death, no one even has a strategy for how to solve his big challenge.
This article was published in print as "Goldbach's Prime Numbers."
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41 Comments
Add Comment"prime numbers (numbers that cannot be evenly divided by any other number except themselves or 1)"
Reply | Report Abuse | Link to thisLike 2.
candide:
Reply | Report Abuse | Link to thisYour point?
Among other things, there is an alternate way to state Goldbach's strong conjecture, about even numbers. If p and q are two primes and p+q = 2n, where n is an integer, then p = n+m, q = n-m, where m is an integer. Then (n+m)(n-m) = n^2 - m^2 = pq. In other words, if n is any integer, n, greater than or equal to 3, there is another integer, m, greater than or equal to 0, such that n^2 - m^2 is the product of two primes.
Reply | Report Abuse | Link to thisThe assertion "coming closer to solution" implies the solution is already known, and progress toward it moves on some kind of highway. Spin?
Reply | Report Abuse | Link to thisWhy would solving the conjecture be useful if an odd number could be the sum of three different primes? At that point it would become useless for encrypting data.
Reply | Report Abuse | Link to thisIt should be mentioned that the article is not correct in stating that, if the weak conjecture is proved, then the strong conjecture is prove. The "argument" presented by the article is that, if every odd numebr can be represented as the sum of three primes, then subtract three and the resulting even number will be the sum of two primes. In fact, that is not guaranteed! The only way subtrating three will automatically make the remainder the sum of two primes is if 3 is one of the three primes summing up to the odd number!
Reply | Report Abuse | Link to thishaha, nice, was that fermat?
Reply | Report Abuse | Link to thisor Euler
Reply | Report Abuse | Link to this57 = 19 + 17 + 13 + 5 + 3
Reply | Report Abuse | Link to thisYeah, that was Fermat on his Last Theorem, scribbled in the margin of something. He said he had discovered a truly marvellous proof of the theorem but the margin was too small to contain it. Given that Andrew Wiles' proof ran to several hundred pages, I guess he was right.
Reply | Report Abuse | Link to thisArticle says if the strong conjecture was true, then the weaker would follow, not how you put it.
Reply | Report Abuse | Link to thisI'll bet his taxes were audited.
Reply | Report Abuse | Link to thisI think you've misunderstood the argument being made. The claim was that the strong conjecture implies the weak one, not the other way around. If it were true that any even number could be written as the sum of two primes, then any odd number could be written as the sum of three primes by just adding three to the two primes summing to the even number that is three less than it.
Reply | Report Abuse | Link to thisUnknown's post, namely 57=19+17+13+5+3, would seem to be a counterexample to the weak Goldbach conjecture (as stated in the article) that any odd number can be broken up into the sum of at most three primes, thus disproving the conjecture. What am I missing?
Reply | Report Abuse | Link to thisThe 57=19+17+13+5+3 is not a counterexample.
Reply | Report Abuse | Link to thisIt's simply a way of writing 57 as the sum of 5 primes.
I counted 16 different ways of writing 57 as the sum of 3 primes, e.g.
57=3+7+47 .... 57=19+19+19
A counterexample to the Weak Goldbach Conjecture would be finding an odd number that cannot be written as the sum of 1, 2 or 3 primes.
The original Wiles' proof was not several hundred pages but over one hundred for sure (I've seen it). It was available in some web sites.
Reply | Report Abuse | Link to thisMany people thought that Fermat was wrong about the comments of his proof. But I think that Fermat might have been right about his "marvelous" proof. I've seen a proof that can be short, but its lemmas take several pages. Note: even high school students could understand it. Conjecture looks so innocent but when it is applied by integer, the equation becomes as false as 1=2.
I must be dumb, but for the life of me I can not see where there is a "mathematical problem" here, it is pure logic.
Reply | Report Abuse | Link to thisAll odd numbers are made up of two primes and all primes can be made up by a maximum of three primes. At least that is how I have always thought of numbers. I am no mathematician, anything but, so what am I missing here?
This is really good news. I noticed the news on Nature website, May 15, 2012.
Reply | Report Abuse | Link to thisI had solved the strong Goldbach’s conjecture completely. This manuscript has been submitted to the journal for publishing. In this paper formulas for primes generate only primes, exactly and without exception. And with one of the applications of these formulas, the Goldbach’s Conjecture proposition: Every even positive integer greater than 2 can be written as the sum of two primes is true.
By the way, the checking program edited with Formulas for Primes by Units’ Digit Sieve Method had checked less than 10,000,000,000 prime numbers are true.
Golbach Conjecture is prooved
Reply | Report Abuse | Link to this2x=(x+r)+(x-r)
A proof of Goldbach and de Polignac conjectures
Jamel Ghanouchi
6 Rue Khansa 2070 Marsa Tunisiere
Proof Link here http://unsolvedproblems.org/S20.pdf
Same proof idea here
Solution for Goldbach Conjecture
Proof Link
vixra.org/pdf/1204.0021v3.pdf
Every number in N could be written as a sum of (n+a)
n+a = 2n+(n-a) = 2n-(n-a)
=> 2n = (n+a)+(n-a)
q.e.d.
Sorry correction
Reply | Report Abuse | Link to thisEvery number in N could be written as a sum of (n+a)
n+a = 2n+(a-n) = 2n-(n-a)
=> 2n = (n+a)+(n-a)
q.e.d.
ABSTRACT. In number theory, if an integer greater than 1 is not a prime number, it must be the composite. All even numbers that are greater than 2 are composites and all odd numbers that are greater than 5, whose units’ digit is 5, are composites. Meanwhile, all prime numbers, except 2 and 5, whose units’ digit must be 1, 3, 7 and 9. In this paper, we describe the formulas for primes whose units’ digit is 1, 3, 7 and 9. As an application of these formulas, we prove that the Goldbach’s Conjecture proposition, viz. every even positive integer greater than 2 can be written as the sum of two primes, is true.
Reply | Report Abuse | Link to thisThis paper has been submitted to AIM. No. AIM-D-12-00347.
Actually, Goldbach's Conjecture (GC) should have a caveat as following: any even number greater than 8 is a sum of two primes. As you said, 1 is not a prime number. Thus 1+3, 1+5 and 1+7 will easily counter GC, if general even number is used. I don't really delve into GC, I would like to see how you prove it.
Reply | Report Abuse | Link to thisLet's look at the following equation:
2n = a + b; n=5, 6, 7,...
IMHO: If the proof assumes that a and b are prime numbers, then all bets are off. It is because 2n can be produced by two non-prime numbers.
I don't know about the application and/or significant of the Strong and Weak Goldbach's Conjecture, it seems to me that it is as useless as Fermat's Last Theorem, other than pure mathematical exercises. But I could be wrong.
The article mentioned that an application may be used in a cryptography. I'd like to see the approach to that encryption scheme.
-Joe
Actually, Formulas for primes are be used in a cryptography.If someone need the paper titled "Formulas for primes and Goldbach's conjecture". Please inform me,
Reply | Report Abuse | Link to thisThis is the e-mail, yrgaodeyao at yahoo dot com dot cn.
57 = 3+23+31 dumbass
Reply | Report Abuse | Link to thisActually the article says the converse is true. Stronger implies weak. It is easy to see why.
Reply | Report Abuse | Link to thisMeanwhile, for those who are interested, the stronger Goldbach conjecture is exceptionally difficult and there is one mathematician (i.e someone who knows enough mathematics to be taken seriously) who has claimed that it may be independent of the set theory axioms that we follow.
Reply | Report Abuse | Link to thisIn other words, it may be true for all natural numbers but still unprovable!
"It should be mentioned that the article is not correct in stating that, if the weak conjecture is proved, then the strong conjecture is prove."
Reply | Report Abuse | Link to thisThe article does not say that. What the article actually *does* say is true.
"The "argument" presented by the article is that, if every odd numebr can be represented as the sum of three primes, then subtract three and the resulting even number will be the sum of two primes. "
No, that is not the argument in the article. The argument in the article is that, if Goldbach's Conjecture is true, then subtracting 3 from any odd number yields an even number to which Goldbach's Conjecture can be applied, and so the odd number can be composed of the the sum of three primes.
"The assertion "coming closer to solution" implies the solution is already known, and progress toward it moves on some kind of highway. Spin?"
Reply | Report Abuse | Link to thisThis comment would only make sense if results in mathematics were completely random, never an outgrowth of previous work. As in all fields of research, mathematical results generally do progress along "some kind of highway", building upon previous work. This is certainly true in number theory.
"Why would solving the conjecture be useful if an odd number could be the sum of three different primes? At that point it would become useless for encrypting data."
Reply | Report Abuse | Link to thisActually, proving the conjecture one way or the other, and even whether the conjecture is true, are completely irrelevant to cryptography, just as whether every integer is the product of two primes (they aren't, of course) is irrelevant ... cryptography is based on those numbers that *are* the product of two primes. If Terence Tao really made the claim that proving the Goldbach Conjecture would have practical consequences, he is seriously confused about the import of mathematical proof ... especially proofs of conjectures that are presumed true.
"Unknown's post, namely 57=19+17+13+5+3, would seem to be a counterexample to the weak Goldbach conjecture (as stated in the article) that any odd number can be broken up into the sum of at most three primes, thus disproving the conjecture. What am I missing?"
Reply | Report Abuse | Link to thisReading comprehension? No where in the article does it say "at most".
"All odd numbers are made up of two primes"
Reply | Report Abuse | Link to thisUh, no, 5 is the only odd number that is the sum of two primes. Goldbach's conjecture is that all *even* numbers are the sum of two primes.
"and all primes can be made up by a maximum of three primes. At least that is how I have always thought of numbers."
How you have always thought of something is not relevant. If you have a proof of the claim, offer it.
"I am no mathematician, anything but, so what am I missing here?"
What *aren't* you missing?
"I must be dumb, but for the life of me I can not see where there is a "mathematical problem" here, it is pure logic."
Reply | Report Abuse | Link to thisLike many people who use the word "logic", you don't understand what it is. Hint 1: how things seem to you is not logic. Hint 2: Mathematical proofs are an application of pure logic.
"Many people thought that Fermat was wrong about the comments of his proof. But I think that Fermat might have been right about his "marvelous" proof."
Reply | Report Abuse | Link to thisAlmost certainly not; the proof requires areas of mathematics that were not even close to having been invented at the time.
"Sievert: Every number in N could be written as a sum of (n+a)"
Reply | Report Abuse | Link to thisSigh. This has nothing to do with Goldbach's Conjecture. No one has ever questioned that every integer can be written as the sum of two other integers, and your "proof" doesn't even prove *that*.
"Meanwhile, all prime numbers, except 2 and 5, whose units’ digit must be 1, 3, 7 and 9."
Reply | Report Abuse | Link to thisThis follows directly from the fact that 2 and 5 are factors of 10, our common number base ... but the choice of number base is ad hoc and irrevelant.
"In this paper, we describe the formulas for primes whose units’ digit is 1, 3, 7 and 9. As an application of these formulas, we prove that the Goldbach’s Conjecture proposition"
Not a chance.
"Actually, Goldbach's Conjecture (GC) should have a caveat as following: any even number greater than 8 is a sum of two primes."
Reply | Report Abuse | Link to thisNo, it shouldn't ... 2, 4, and 6 are all the sum of two primes.
"As you said, 1 is not a prime number. Thus 1+3, 1+5 and 1+7 will easily counter GC,"
Complete and utter logic fail ... a fallacy of denial of the antecedent. (Or is it affirmation of the consequent? Well, they're equivalent.)
The mathematical illiteracy of most of the people posting here is somewhat tragic.
"2n = a + b; n=5, 6, 7,...
Reply | Report Abuse | Link to thisIMHO: If the proof assumes that a and b are prime numbers, then all bets are off. It is because 2n can be produced by two non-prime numbers."
Your HO is irrelevant. If a "proof" assumes that a and b are prime numbers then it is circular, since that is the assertion to be proved. That 2n is the sum of two non-prime numbers is completely irrelevant.
" it may be independent of the set theory axioms that we follow.
Reply | Report Abuse | Link to thisIn other words, it may be true for all natural numbers but still unprovable!"
Those are not other words for the same thing. We know from Gödel that, if axioms of arithmetic are consistent, there are true statements for which there is no proof procedure based solely on those axioms. This has nothing to do with being independent of the axioms. And the set theory axioms aren't relevant here, Peano's axioms are.
""As you said, 1 is not a prime number. Thus 1+3, 1+5 and 1+7 will easily counter GC"
Reply | Report Abuse | Link to thisWhy did you stop at 8? 1+9, 2+8, 4+6 are all sums of two numbers that are not both prime. And if the pattern is one prime and one non-prime, we have 1+p for all primes.
I think there's something to be learned here, which is that articles such as this one are mostly being read by people unable to comprehend them.
"'All odd numbers are made up of two primes'
Reply | Report Abuse | Link to thisUh, no, 5 is the only odd number that is the sum of two primes."
Oops, sorry about that ... p+2 for all primes p > 2 is an odd number that is the sum of two primes, and there are obviously infinitely many of those (Euclid proved it), but this has nothing to do with Goldbach's Conjecture, and the statement that all odd numbers are the sum of two primes has an infinite number of exceptions ... n+2 for all odd composite n, e.g., 11, 17, 23, 27, etc.
"The mathematical illiteracy of most of the people posting here is somewhat tragic."
Reply | Report Abuse | Link to thisPerhaps that would include the person you're looking in the mirror! Especially when you made the following comments: "No, it shouldn't ... 2, 4, and 6 are all the sum of two primes."
0+2=2; 1+1=2; Neither 0 nor 1, as we know them, are prime.
I agree that if you count 2+2=4 and 3+3=6, where two and three are each counted twice. Thus definition of GC must be made clearer. IMHO.