# Goldbach conjecture...so what? **URL:** \ **Category:** Factual Questions\ **Created:** [March 13, 2001, 6:23pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819 "2001-03-13T18:23:39Z")\ **Posts on this page:** 15\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [March 13, 2001, 6:23pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/1 "2001-03-13T18:23:39Z")
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? ---
**Author:** ![\_Tim](https://avatars.discourse-cdn.com/v4/letter/_/e47c2d/32.png) [@\_Tim](https://boards.straightdope.com/u/_Tim)\ **Post date:** [March 13, 2001, 6:49pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/2 "2001-03-13T18:49:09Z")
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? ---
**Author:** ![jmullaney](https://avatars.discourse-cdn.com/v4/letter/j/85e7bf/32.png) [@jmullaney](https://boards.straightdope.com/u/jmullaney)\ **Post date:** [March 13, 2001, 6:58pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/3 "2001-03-13T18:58:11Z")
From the britannica: > [@](#): > > Goldbach first proposed the conjecture that bears his name in a letter to the Swiss > mathematician Leonhard Euler in 1742. He claimed that “every number greater than 2 > is an aggregate of three prime numbers.” Because mathematicians in Goldbach’s day > considered 1 a prime number (prime numbers are now defined as those positive > integers greater than 1 that are divisible only by 1 and themselves), Goldbach’s > conjecture is usually restated in modern terms as: Every even natural number greater > than 2 is equal to the sum of two prime numbers. As for the OP – why does a mountain climber climb a mountain? ---
**Author:** ![bibliophage](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bibliophage/32/7615_2.png) [@bibliophage](https://boards.straightdope.com/u/bibliophage)\ **Post date:** [March 13, 2001, 8:04pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/4 "2001-03-13T20:04:46Z")
> [@](#): > > this is apparently the only major math problem left unsolved 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-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html) [http://www.math.purdue.edu/~eremenko/uns.html](http://www.math.purdue.edu/~eremenko/uns.html) ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 13, 2001, 8:24pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/5 "2001-03-13T20:24:51Z")
> [@](#): > > \*Originally posted by enolancooper \* > \*\*What’s the big deal about Goldbach? \*\* 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](http://www.cs.unb.ca/~alopez-o/math-faq/node61.html). Hmm. We had a thread a month or two ago about tough math problems, didn’t we? ---
**Author:** ![don\_willard](https://avatars.discourse-cdn.com/v4/letter/d/a88e57/32.png) [@don\_willard](https://boards.straightdope.com/u/don_willard)\ **Post date:** [March 13, 2001, 9:50pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/6 "2001-03-13T21:50:27Z")
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. ---
**Author:** ![Joe\_Cool](https://avatars.discourse-cdn.com/v4/letter/j/ecb155/32.png) [@Joe\_Cool](https://boards.straightdope.com/u/Joe_Cool)\ **Post date:** [March 13, 2001, 10:13pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/7 "2001-03-13T22:13:33Z")
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](http://pass.maths.org.uk/issue2/xfile/index.html), Goldbach’s Conjecture is this: > [@](#): > > Every even number greater than 4 can be written as the sum of two odd prime numbers. [This site](http://www.newton.dep.anl.gov/askasci/math98.htm) (item #98), however, says this: > [@](#): > > Christian Goldbach (1690-1764), in a letter to Euler dated February, 16, 1745, stated that every even number equal to or greater than 6 is the sum of two odd primes in one or more ways. The first sentence is a direct quote from THE LORE OF PRIME NUMBERS by G. P. Loweke, p 68. …While [_this_ site](http://www.utm.edu/research/primes/notes/conjectures/) says: > [@](#): > > Goldbach wrote a letter to Euler in 1742 suggesting that every integer n \> 5 is the sum of three primes. Euler replied that this is equivalent to every even n \> 2 is the sum of two primes–this is now know as Goldbach’s conjecture. Schnizel showed that Goldbach’s conjecture is equivalent to every integer n \> 17 is the sum of three distinct primes. …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. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [March 13, 2001, 10:13pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/8 "2001-03-13T22:13:35Z")
I’d say the biggest unsolved math problem right now is the [Riemann Hypothesis](http://www.utm.edu/research/primes/notes/rh.html). 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. ---
**Author:** ![Ben](https://avatars.discourse-cdn.com/v4/letter/b/df788c/32.png) [@Ben](https://boards.straightdope.com/u/Ben)\ **Post date:** [March 13, 2001, 11:12pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/9 "2001-03-13T23:12:13Z")
> [@](#): > > \*Originally posted by Joe\_Cool \* > \*\* > 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. \*\* 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 ---
**Author:** ![jcgmoi](https://avatars.discourse-cdn.com/v4/letter/j/a9adbd/32.png) [@jcgmoi](https://boards.straightdope.com/u/jcgmoi)\ **Post date:** [March 14, 2001, 12:17am UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/10 "2001-03-14T00:17:52Z")
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: > [@](#): > > I’d say the biggest unsolved math problem right now is the Riemann Hypothesis. No argument. Maybe that’s what **HoldenCaulfield** has resolved. ---
**Author:** ![Joe\_Cool](https://avatars.discourse-cdn.com/v4/letter/j/ecb155/32.png) [@Joe\_Cool](https://boards.straightdope.com/u/Joe_Cool)\ **Post date:** [March 14, 2001, 3:00am UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/11 "2001-03-14T03:00:47Z")
> [@](#): > > \*Originally posted by Ben \* > \*\*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 \*\* 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. ---
**Author:** ![knappy](https://avatars.discourse-cdn.com/v4/letter/k/b3f665/32.png) [@knappy](https://boards.straightdope.com/u/knappy)\ **Post date:** [March 14, 2001, 2:02pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/12 "2001-03-14T14:02:17Z")
don willard: > [@](#): > > 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 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. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 14, 2001, 5:06pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/13 "2001-03-14T17:06:40Z")
> [@](#): > > \*Originally posted by don willard \* > \*\*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. \*\* 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. ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [March 14, 2001, 9:39pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/14 "2001-03-14T21:39:10Z")
> [@](#): > > \*Originally posted by don willard \* > \*\* IS IT REALLY IMPOSSIBLE TO STATE MATHEMATICAL PROBLEMS SO THAT PEOPLE WITH AVERAGE INTELLIGENCE CAN UNDERSTAND THEM? > \*\* Depending on what you mean by “average”, of course it isn’t impossible. Authors like [Hofstadter](http://www.amazon.com/exec/obidos/ASIN/0465026567/qid=984605015/sr=1-1/ref=sc_b_2/106-1989989-9480411) and [Dunham](http://www.amazon.com/exec/obidos/ASIN/014014739X/qid%3D984605104/106-1989989-9480411) 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. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [March 14, 2001, 11:01pm UTC](https://boards.straightdope.com/t/goldbach-conjecture-so-what/57819/15 "2001-03-14T23:01:39Z")
**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.