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May 20

Goldbach conjecture

Can someone explain why this doesn't solve the Goldbach conjecture? I know nothing of maths.

"1 - All of the prime numbers other than 2 are odd, 2 being the only even prime number. Further, the even number 4 = 2 + 2.

2 - The sum of any two of the odd prime numbers is always an even number.

3 - All combinations* of the odd numbers ≥ 3 [whether prime or not] summed in pairs produces all of the even numbers ≥ 6.

3 - While just the prime odd numbers in sequence is a sequence with gaps as compared to that of all of the odd numbers; nevertheless, all combinations of the odd prime numbers ≥ 3 summed in pairs produces all of the even numbers provided that there are enough primes preceding the gaps.

4 - That requirement is that π(N) ≥ R(N) = G·Ln(N) where N is the first number in the gap, π(N) is the number of primes less than or equal to N, R(N) is the number of preceding primes needed to assure clearance of the gap, and G is the number of sequential non-primes in the gap. This requirement is comprehensively satisfied by all of the prime numbers and gaps because of the sufficiently smooth nature of π(N)." 86.8.176.85 (talk) 02:00, 20 May 2009 (UTC)

Do you care to tell us where you found this? —Tamfang (talk) 06:34, 20 May 2009 (UTC)

most likely here: [1] --pma (talk) 06:57, 20 May 2009 (UTC)

A friend of mine who does maths as a hobby was explaining the Goldbach conjecture to me, and told me he was sure this doesn't solve it but unsure why. Just wondering if anyone could tell me what's wrong with it... 86.8.176.85 (talk) 19:03, 20 May 2009 (UTC)

This is basically an argument that Goldbach's conjecture is plausible because there are "enough primes" available to make sums from. The primes look randomly distributed, and if you pick a bunch of random odd numbers distributed similarly to the primes then it's exceedingly unlikely that there will be any large even number you can't make with them (and the small even numbers have been tested). Goldbach's conjecture#Heuristic justification has more details. To prove the conjecture you would have to completely rule out the alternative, not just judge it implausible. The plausibility argument is obvious to modern mathematicians, so the author of this paper hasn't done anything new. I don't know how obvious it was in 1742 when Goldbach made the conjecture. -- BenRG (talk) 19:40, 20 May 2009 (UTC)

The term "plausibility argument" is standard for this sort of thing, but in cases like the Goldbach conjecture it rather understates the case.

The Goldbach conjecture is true. It's a mortal lock. In a corresponding case in the physical sciences we would have no squeamishness about expressing it that way. It's only the availability of proof-from-axioms in mathematics, that makes us worry about the fact that we don't have one (from any widely accepted axiom set) in the case of Goldbach. --Trovatore (talk) 20:33, 20 May 2009 (UTC)

No, this is wrong. The physical sciences don't deal with philosophical absolute truth. Math does. And we do have conjectures that check out for extremely large sets of numbers, but do eventually yield under an even larger counterexample (for a trivial example, use "all integers are smaller than 10^{((((((((((((10!)!)!)!)!)!)!)!)!)!)!)!)!}. --Stephan Schulz (talk) 08:49, 21 May 2009 (UTC)

The notion that mathematics deals in apodeictic certainty is an error that goes back to the ancients, but an error nonetheless. The differences between mathematics and the empirical sciences is one of degree, not kind. You always need axioms, and the axioms could be wrong.

If you think that the argument is simply that Goldbach holds up to very large values, then you haven't understood the argument. Review the link Ben provides above. --Trovatore (talk) 09:56, 21 May 2009 (UTC)

Can you explain in what sense axioms can be "wrong"? One way is obviously that they're inconsistent. Another would be that they do not correctly describe some physical reality (like Euclidean geometry vs. relativistic space), but I think we all agree that a physical interpretation is not a requirement.

I have always been under the impression that mathematics aspires to make certain deductions. While the statement "π is irrational" might not be an absolute truth, the statement "from the ZFC axioms it follows that π is irrational" should be. If ZFC turns out to be inconsistent, the latter will still be an absolute truth, as will "from the ZFC axioms it follows that π is rational".

Another issue that may be relevant here is that even if some statement is true, we humans will never be able to know this with certainty. Usually we are convinced when we see a proof, but it's possible the proposed proof contains a mistake which we have overlooked. We can overlook it again when we re-read the proof, and it's possible (even if extremely implausible) that every human or computer that has ever checked the proof has overlooked it. -- Meni Rosenfeld (talk) 11:10, 21 May 2009 (UTC)

Obviously I can't do justice here to the realist–formalist debate. I'll limit myself to noting the following:

• Most people are realists about at least small natural numbers; they think of the number 2 as being a real, though non-physical, object, and if a collection of axioms proves that 2+1=5, then at least one of those axioms is wrong (assuming that 2,1,5, and + are expected to have their usual interpretations).
• The existence of a formal, syntactic proof of a formal sentence from ZFC, assuming it's short enough to actually write down, is indeed about as absolute as things get (maybe less so than cogito ergo sum). However it's not very interesting. It gets more interesting if you interpret it semantically, as "this statement is true in every model of ZFC". But only if you're willing to make an ontological commitment to models of ZFC, and in that case you're back to realism. (By the way, this sort of "realism about models" is enough to guarantee that CH has a determinate truth value, at least given that the powerset axiom is actually true, which is itself a claim that a realist-about-models must consider well-specified.)
• I disagree that the essence of mathematics is to make deductions. The essential purpose of mathematics is to discover what the truths are about mathematical objects. In the case of the Goldbach conjecture, we know the truth, at least if knowledge is interpreted as justified true belief. --Trovatore (talk) 23:34, 21 May 2009 (UTC)

While I find very instructive your point of view, your statement about the Goldbach conjecture seems a bit optimistic to me. For sure we think that it is true, but it seems to me that this belief is based on the fact that all that it is known about the problem suggests a very reasonable statistic model where this conjecture is true, with extremely high probability. So in a certain sense this belief is based on our lack of knowledge of the problem. Don't you think that a new result may change our state of knowledge and force to abandon that model (in particular, the independence assumptions, that are based on a principle of indifference)? --pma (talk) 01:26, 22 May 2009 (UTC)

Oh, it's conceivable in principle, certainly. But that's my point — this is not different, in principle, from our other sorts of mathematical knowledge. It's also conceivable in principle that a new result could change our state of knowledge and cause us to reevaluate some of our axioms. --Trovatore (talk) 01:52, 22 May 2009 (UTC)

No doubt that this is a point of view that one can not ignore. The similarity with physics is very attractive indeed. But my concern is mainly on the status of certainty of Goldbach conjecture in particular. So, if one estimates the number of ways an even number n writes as sum of 2 primes, and one assumes "enough independence" (in some technical sense) about the distribution of primes, one finds that the number of ways is even divergent. This independence is a very reasonable assumption, because there is not a single result, nor any similar conjecture, that may suggest the contrary. As a consequence, we strongly believe that the CG is true. But does it means that we should believe with the same certainty that nobody will disprove it in the next years, or just find a good reason to change the above statistical model? I feel a bit dumb, but I am inclined to say no. The CG has very high probability to be true, given the current state of art, but it seems to me vaguely improper to argue from this, that any new change of the state of art is most likely one going in the direction of the proof. After all, what mathematicians usually do, is to find unexpected results, able to completely change a perspective. If today we look at primes like a "swarm of bees", it is maybe just because they have not yet done the next step into the knowledge.--pma (talk) 14:17, 22 May 2009 (UTC)

As to the probabilistic argument about the validity of Goldbach conjecture, let's recall that although very meaningful, it is far from an absolute statement. As I see it, it says that, given our current knowledge about prime numbers, it is extremely unlikely that a counterexample exists. So, it is rather a description of the current state of knowledge of the problem, than a definitive claim on the trueness of the GC. Indeed, it is based upon a model that may be completely changed after a single observation --of course, that would be ipso facto a very deep one. (To make an example, there was a time when the knowledge of geometry was such, to make very plausible the conjecture that the edge and the diagonal of a square be in rational ratio. Why to assume there is a non-rational ratio, if all you know is rational?). Indeed I do not see how the probability that a given conjecture is true may have a meaning outside a relative, Bayesian point of view.

Moreover, the fact is that we are more curious to see a proof of a mathematical fact, than to know whether it is true. Suppose the archangel Michael appears to a number theorist claiming that GC he's trying to prove is true. Then, he would be a bit embarassed... while certainly glad of the visit, I don't think he would give up working on it. --pma (talk) 22:47, 21 May 2009 (UTC)

If I remember correctly, a so-far-unwritten article on A.I.Vinogradov would have to have partial results, rather than the layperson material about mathematical proof that has been discussed here.Julzes (talk) 06:48, 27 May 2009 (UTC)

Parimutuel betting

As a novice, I have been trying to understand the mathematics of parimutual betting. I can understand the (theorectical) situation where bettors are only allowed to bet on the winner. After deduction of the running costs, the total money bet by all punters is divided up among the winning bettors in proportion to how much they have bet.

But how is the money divided up when other types of bet are also allowed? The win, place, show, and more exotic bets? 89.242.109.25 (talk) 12:21, 20 May 2009 (UTC)

I imagine you would create a separate pool for each type of bet (win, place, show etc.), pro-rata costs across pools (i.e. take same percentage cut from each pool), then divide net amount left in each pool between the folks who placed winning bets of that type. I can see a problem with more exotic bet types though - what happens to the trifecta pool, for example, if no-one correctly predicts first, second and third places ? Gandalf61 (talk) 14:43, 20 May 2009 (UTC)

From reading the article (and the talk page), I get the impression that separate pools for each bet type are typically used. However another possibility (perhaps never employed) is to have a single pool, and then fix the ratio of the payouts. For example, the winners on the "win" bet could receive 3 times that of any "place" bet winners, who receive twice that of any "show" bet winners. From there it's simple algebra to figure out the payouts for each bet. -- 128.104.112.117 (talk) 18:59, 24 May 2009 (UTC)

transcendental equation

May somebody help me solve this equation? (Maple's numerical methods don't work, apparently) --Taraborn (talk) 14:03, 20 May 2009 (UTC)

sin(7.147*sin(x)) / (7.147*sin(x) = 1/sqrt(2)

And this one:

cos(x)*sin(9.3462*sin(x)) / (9.3462*sin(x)) = 1/sqrt(2)

Thanks. --Taraborn (talk) 14:03, 20 May 2009 (UTC)

I get 0.195957 and 0.147234 for the smallest positive solutions; in each case the function is even, of course, and they have periods π and 2π. Is that what you needed? --Tardis (talk) 14:24, 20 May 2009 (UTC)

No, both equations don't have any solutions at all. You can find this by plotting the function and checking that they don't intersect where they should for a solution to exist. - DSachan (talk) 19:36, 20 May 2009 (UTC)

Erm, it looks to me like each of those equations has solutions, where Tardis said. There is no simultaneous solution to both equations, but separately each one is OK. —Bkell (talk) 21:56, 20 May 2009 (UTC)

What are the solutions then? The solutions given by Tardis are not correct. Alright, sorry, everything seems to be alright. I goofed up a bit. Solutions given by Tardis are correct ones. - DSachan (talk) 22:29, 20 May 2009 (UTC)

does anyone know how to solve this????

(18 + 308^1/2)^1/2

this equation can be written in the form of a^1/2 + b^1/2, where a and b are whole numbers and a is greater than b.

i need the values of a and b.

plz hhhhheeeeelllllpppp.... —Preceding unsigned comment added by 117.197.245.126 (talk) 14:49, 20 May 2009 (UTC)

As far as I know how it works here, people will help you sooner, if you show that you have made some effort.

I write the equation for you:

(18 + 308^1/2)^1/2 = a^1/2 + b^1/2

Now, what is your next step, to find a and b? (the funny thing is that a and b are hidden in your user number) --84.221.208.46 (talk) 14:57, 20 May 2009 (UTC)

It may help to see if you can put √308 into the form c√d, where c,d are whole numbers. —Tamfang (talk) 15:20, 20 May 2009 (UTC)

ok, i've done it till (18 + 2(77)^1/2)^1/2 now what???? —Preceding unsigned comment added by 122.50.130.33 (talk) 16:15, 20 May 2009 (UTC)

Well, what you want is

(18 + 308^1/2)^1/2 = a^1/2 + b^1/2

that is, if you square both sides,

18 + 308^1/2 = (a^1/2 + b^1/2)².

So why don't you try expanding the square at the RHS,

(a^1/2 + b^1/2)² = a + b + 2(ab)^1/2,

and then try writing two equations:

18 = integer term on the LHS = integer term on the RHS = a + b;

308^1/2 = square root on the LHS = square root on the RHS = 2(ab)^1/2.

Then get rid of the square root in the second equation. Try it, it's not that difficult; you'll get

a + b = 18

ab = 77.

Now either you find the solution at the first glance (remember that a and b are integer numbers), or you first compute as usual

(a - b)² = (a + b)² - 4ab.

84.221.208.46 (talk) 16:35, 20 May 2009 (UTC)