Editing 1310: Goldbach Conjectures
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==Explanation== | ==Explanation== | ||
− | + | {{w|Goldbach's conjecture}} and the {{w|Twin prime|twin prime conjecture}} are unsolved problems in mathematics relating to {{w|prime numbers}} (numbers whose only {{w|divisors}} are 1 and itself). A claimed proof of {{w|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. | |
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− | + | 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<sup>14348907</sup>. 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 10<sup>30</sup>, while all numbers below 10<sup>30</sup> 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. | |
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− | + | 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. | |
+ | ---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 between 6 and 7. Also, the extremely weak conjecture could easily refer to numbers in the negative direction only. | ||
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− | + | 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. | |
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− | + | [[Randall]]'s weak twin prime conjecture states that there are an infinite number of pairs of primes. This is clearly true. Per {{w|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. | |
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+ | 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 {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first"). | ||
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+ | The tautological prime conjecture states that it itself is true, while making no statement about primes. It is not technically a {{w|tautology}} but more of a plain assertion. Randall has mentioned tautologies before in [[703: Honor Societies]]. | ||
==Transcript== | ==Transcript== | ||
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:'''Goldbach Conjectures''' | :'''Goldbach Conjectures''' | ||
− | + | :'''Weak''' | |
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− | :'''Weak | ||
:Every odd number greater than 5 is the sum of three primes | :Every odd number greater than 5 is the sum of three primes | ||
− | + | :'''Strong''' | |
− | :'''Strong | ||
:Every even number greater than 2 is the sum of two primes | :Every even number greater than 2 is the sum of two primes | ||
− | + | :'''Very weak''' | |
− | :'''Very strong | + | :Every number greater than 7 is the sum of two other numbers |
+ | :'''Very strong''' | ||
:Every odd number is prime | :Every odd number is prime | ||
− | + | :'''Extremely weak''' | |
− | :'''Extremely strong | + | :Numbers just ''keep going'' |
+ | :'''Extremely strong''' | ||
:There are no numbers above 7 | :There are no numbers above 7 | ||
{{comic discussion}} | {{comic discussion}} | ||
− | [[Category: | + | [[Category:Math]] |
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