1310: Goldbach Conjectures

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Goldbach Conjectures

Title text: The weak twin primes conjecture states that there are infinitely many pairs of primes. The strong twin primes conjecture states that every prime p has a twin prime (p+2), although (p+2) may not look prime at first. The tautological prime conjecture states that the tautological prime conjecture is true.

Explanation

Goldbach's conjecture and the twin prime conjecture are unsolved problems in mathematics relating to prime numbers (numbers greater than 1 whose only divisors are 1 and itself). A claimed proof of Goldbach's weak conjecture is currently under review.

Randall is riffing on the relationship between "strong" and "weak" logical statements, which are an interplay between the boldness or usefulness of a statement and the ease with which it might be proven to be true. For example, if Goldbach's conjecture (given in the comic under the label "strong") could be proven to be true, it would automatically imply that Goldbach's weak conjecture (given in the comic under the label "weak") is also true, because any odd number greater than 5 can be expressed as 3 (a prime number) plus an even number greater than 2 (which, per the strong conjecture, would itself be the sum of two prime numbers), resulting in a way to express the original odd number as the sum of three prime numbers. The weak conjecture does not, however, imply the strong conjecture.

Mathematicians have been solving related problems that are "weaker" than the weak conjecture, and working towards "stronger" ones. For example, in 1937 the weak conjecture was proven for odd numbers greater than 3^14348907. In 1995 a version was proven based on the sum of no more than seven prime numbers, and in 2012 the ceiling was lowered to five primes. In 2013 the weak conjecture was claimed proven for numbers greater than 10³⁰, while all numbers below 10³⁰ have been verified by supercomputer to satisfy the conjecture; these together imply that the weak conjecture is true (although there is no general proof of it for all numbers). Goldbach's strong conjecture remains unsolved.

This comic plays on the "strong" and "weak" naming of Goldbach's conjectures by extending it to further degrees of strength or weakness. The "very weak" and "extremely weak" conjectures are indeed implied by Goldbach's weak conjecture, just as the weak conjecture is implied by the strong one. The "very strong" and "extremely strong" conjectures are extensions of Goldbach's strong conjecture, even as it is an extension of the weak conjecture. However, the "very weak" and "extremely weak" conjectures are so obviously true that they are hardly worth stating, while the "very strong" and "extremely strong" conjectures make such bold claims that they are obviously false.

Moreover, the "extremely weak" and "extremely strong" conjectures contradict each other, even though they're both derived (albeit in opposite directions) from the same initial conjectures. Therefore, the "extremely strong" conjecture could not possibly imply (however indirectly) the validity of the "extremely weak" conjecture, as it would if proved true.

The title text refers to the twin prime conjecture, which states that there are an infinite number of pairs of primes that differ by 2, and then applies the same spectrum of "weak" and "strong" statements to it.

Randall's weak twin prime conjecture states that there are an infinite number of pairs of primes. This is clearly true. Per Euclid's theorem, there are an infinite number of primes. Unlike the actual twin prime conjecture (which specifies a distance of two), this conjecture does not specify a required distance. Thus, any pair from the infinite set of primes suffices. An example is 5 and 13.

His strong twin prime conjecture states that every prime is 2 less than another prime. This statement is obviously false, as there are many possible counter-examples to this statement (thus Randall's humorous hedge that some prime numbers "may not look prime at first").

The tautological prime conjecture states that it itself is true, while making no statement about primes. It is not technically a tautology but more of a plain assertion. Randall has mentioned tautologies before in 703: Honor Societies.

Transcript

Goldbach Conjectures

Weak

Every odd number greater than 5 is the sum of three primes

Strong

Every even number greater than 2 is the sum of two primes

Very weak

Every number greater than 7 is the sum of two other numbers

Very strong

Every odd number is prime

Extremely weak

Numbers just keep going

Extremely strong

There are no numbers above 7

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Discussion

If a bot can create the text I read here, we have made great strides in artificial intelligence. Probably a human editor forgot to change the "incomplete/incorrect" heading. Tenrek (talk) 05:53, 30 December 2013 (UTC)

You never know, AI has come a loong way. Davidy²²[talk] 06:39, 30 December 2013 (UTC)

Let's ask: Tepples, are you a bot? And 199.27.128.62, what about you? -- Hkmaly (talk) 10:09, 30 December 2013 (UTC)

Yes, I'm a bot. 199.27.128.62 21:42, 30 December 2013 (UTC)

I thought that {{incomplete|Created by a BOT}} means that the template was inserted by a BOT. 173.245.50.84 13:55, 30 December 2013 (UTC)

It does mean that. But as others edit the page, they should keep the "incomplete" reason up-to-date. I've changed it to "incomplete|surely not quite complete yet..." ;) Nealmcb (talk) 14:28, 30 December 2013 (UTC)

I will change this text template beginning at the Friday update when I'm back home. Happy NEW YEAR to everybody! --Dgbrt (talk) 15:16, 30 December 2013 (UTC)

It all seems to work except that the extremely strong seems to imply the opposite of the extremely weak Djbrasier (talk) 02:19, 31 December 2013 (UTC)

I think the mistake is in the implication of the very weak to the extremely weak version. In fact, if there is any connection between those two statements it is an implication that goes the other way round. If the extremely strong version is true, we are not looking at the natural numbers. Thus, "Every number greater than 7 is the sum of two other numbers." does not imply "Numbers just keep going.", at all. (Also this accounts for no numbers at all, so the very weak version would still be correct.) Then there is the case that the extremely strong version is false. An implication from something false to anything is always true. --173.245.53.200 07:30, 1 January 2014 (UTC)

---I disagree with this, as it is not incorrect to say that "numbers keep going towards seven" as there are an infinite number of numbers approaching 7. Also, the extremely weak conjecture could easily refer to numbers in the negative direction only. 173.245.54.61 (talk) (please sign your comments with ~~~~)

I always find it amusing that people assume that something phrased 'scientifically' is therefore right, whereas something phrased unscientifically (eg religious beliefs taken on faith) are automatically wrong. There seems to be an unexamined assumption that science is some magical dark art for uncovering infallible truths. Of course science is really just a methodological system for testing theories. Whenever I try to explain this concept, I try to come up with a general, untestable (non-scientific) assertion that is nonetheless true, alongside a very specific, repeatedly testable (falsifiable) assertion that is therefore eminently scientific, but which happens to be wrong. (Eg "it sometimes rains on Wednesday" and "it rains at least 100mm every Wednesday in Riyadh"). So for me this comic is a commentary on that principle - that the "strength" of a statement is only really impressive if it has also survived testing. Tarkov (talk) 10:47, 31 December 2013 (UTC)

The assumption is not "that science is some magical dark art for uncovering infallible truths" but that science works. Bitches. Also, the example you have given is quite bad considering that your first statement is so vague that it is essentially meaningless and apparently, what you want to say with your second statement is that falsifiable claims are falsifiable, which is pretty trivial. Finally, the statements that are phrased unscientifically are not assumed to be automatically wrong but they are impossible to be proven or disproven and are often worded so vaguely that nobody in the known universe knows just what the hell they are supposed to even mean. They are just empty phrases that carry no information whatsoever. --173.245.53.200 07:30, 1 January 2014 (UTC)

According to the strong twin prime conjecture, all positive numbers greater than one are prime, due to 2 and 3 both being prime and extrapolation on primes from there. Thus, this nearly proves the very strong Goldbach conjecture, excluding one. Should this be noted in the explanation? 108.162.237.4 02:08, 1 January 2014 (UTC)(Kyt)

I don't know if it's worth complicating things to bring the matter up. It's potentially more complicated than a simple error; in Goldbach's day, people still sometimes thought of 1 as a prime number (which simplifies his conjectures). —TobyBartels (talk) 18:00, 1 January 2014 (UTC)

This also reminds me of those psychological tests that ask how you feel about this and that. 108.162.226.228 15:02, 1 January 2014 (UTC)

Don't forget the first rule of tautology club. --141.101.98.236 18:07, 1 January 2014 (UTC)

Moved from explain

I disagree with this, as it is not incorrect to say that "numbers keep going towards seven" as there are an infinite number of numbers approaching 7. Also, the extremely weak conjecture could easily refer to numbers in the negative direction only. (Edited by some people.) --Dgbrt (talk) 18:18, 10 January 2014 (UTC)

"Therefore, the "extremely strong" conjecture could not possibly imply (however indirectly) the validity of the "extremely weak" conjecture, as it would if proved true."

It can be argued that since the "extremely strong" conjecture is obviously a contradiction (as in the logical sense, "a formula that's always false"), thereby, can imply any other formula. That is, if p is always false, then (p->q) for any q is always true. In this sense, if the "strong" version gets proved somehow, you get an inconsistent logical system, in which each and every formula can be proved as true, including those weaker forms. 108.162.215.56 13:03, 9 February 2014 (UTC)

This sentence is problematic: "The weak conjecture does not, however, imply the strong conjecture." "A does not imply B" technically means "A and not B" which, I'm sure, isn't what was meant. I added "in any evident way" which I think corrects it. 199.27.133.59 08:52, 10 July 2014 (UTC)

The paradoxical prime conjecture states that the paradoxical prime conjecture is false. --108.162.250.220 07:58, 12 October 2014 (UTC)

KYT's conjecture - prime numbers pattern

KYT's conjecture is described as follows:

a is a positive integer and is even, a>=8, b=a+18, a=c+D, c, D,E are prime numbers.

a=c+D b=c+E

E=D+18=b-c

There must be a prime number c that satisfies the two equations above. More examples are as follows:

10=3+7;5+5 28=5+23;11+17 46=3+43;5+41;17+29;23+23 64=3+61;5+59;11+53 ;17+47;23+41 82=3+79;11+71 ;23+59;29+53;41+41

---

Hello, I am not a regular user here (I have no account), but I am a working analytic number theorist. Harald Helfgott's proof of the ternary (i.e., weak) Goldbach conjecture has indeed been accepted as correct.

162.158.126.204 12:26, 27 August 2024 (UTC) Frank Thorne