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| | ==Explanation== | | ==Explanation== |
| − | In mathematics, a pair of related conjectures may be described as "strong" and "weak" (or often, a normal statement and a "weak" one). A strong conjecture, if true, can be used to easily prove the weaker one, but not vice versa (i.e. if the weak statement is true, that alone isn't enough to prove that the strong one is also true). Conversely, if the weak conjecture is false, that is enough to prove the stronger one false as well, but not vice versa. Weak conjectures are often easier to prove than related strong ones.
| + | {{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. |
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| − | Goldbach's {{w|Goldbach's weak conjecture|weak}} and {{w|Goldbach's conjecture|strong}} conjectures are a pair of real, unsolved problems relating to {{w|prime number}}s (a number with exactly two positive divisors, 1 and itself). The comic states these under the labels "weak" and "strong".
| + | 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. |
| − | * Goldbach's weak conjecture says that every odd number above 5 can be written as the sum of three prime numbers. A computer-aided proof of this was completed in 2013, but it is not yet clear whether the proof has been accepted as correct.
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| − | * Goldbach's strong conjecture (more often, simply "Goldbach's conjecture") says that every even number above 2 can be written as the sum of two prime numbers. If true, this would automatically make the weak conjecture true as well, because every odd number above 5 can be written as an even number above 2 (equal to two primes), plus 3 (the third prime).
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| − | Randall's further conjectures extend this to a whole series of progressively "weaker" and "stronger" statements. His weak conjectures are so weak that they are obviously true; his strong conjectures are so restrictive that they are obviously false. However, for the most part, they really do maintain a weak-strong relationship.
| + | 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. |
| − | * The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.{{citation needed}} But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).
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| − | * The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)
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| − | * In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true,{{citation needed}} but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).
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| − | * The "extremely weak" conjecture says that "numbers just keep going". This is true, but it may not actually be implied by the above conjectures. Those say that numbers above 7 have certain properties, without ''requiring'' that such numbers exist. This may seem like a nitpicky point, but mathematicians love those; it also causes problems, because the "extremely strong" and "extremely weak" conjectures contradict each other. If the other conjectures were rewritten to say "these numbers exist, ''and'' have these properties", then they would imply this "extremely weak" conjecture, but then the "extremely strong" one would have to be stricken off.
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| − | The title text gives the same treatment to the {{w|Twin prime|twin prime conjecture}}, which says that there are infinitely many pairs of primes ''where one is 2 more than the other'' (e.g. 3 and 5). The title text adds a "weak" conjecture, according to which there are simply infinitely many pairs of primes (with no mention of the distance between them). This is true; {{w|Euclid's theorem}} says that there are an infinite number of primes, and so you can simply pick any two (e.g. 5 and 13) and call them a pair.
| + | 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. |
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| − | It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime, for example, but cannot be one of a pair as neither 21 nor 25 are. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first".
| + | 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. |
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| − | Lastly, 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]].
| + | 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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| − | ===History of Goldbach's conjecture===
| + | [[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. |
| − | Mathematician {{w|Christian Goldbach}} wrote a form of his conjecture (the "strong" one of the comic) in a letter to the famous {{w|Leonhard Euler}} in 1742. Euler replied that he considered it certainly true, but that he could not prove it.
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| − | Mathematicians have been solving related problems that are "weaker" than Goldbach's 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 supercomputers 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.
| + | 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]]. |
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| | ==Transcript== | | ==Transcript== |
| − | :[Six small panels with captions are arranged in an arch shape:]
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| − | :[Caption under the arch:]
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| | :'''Goldbach Conjectures''' | | :'''Goldbach Conjectures''' |
| − | | + | :'''Weak''' |
| − | :[Captions in the panels, from left to right:]
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| − | :'''Extremely weak:'''
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| − | :Numbers just ''keep going''
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| − | | |
| − | :'''Very weak:'''
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| − | :Every number greater than 7 is the sum of two other numbers
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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 |
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| | {{comic discussion}} | | {{comic discussion}} |
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| − | [[Category:Charts]] | + | [[Category:Math]] |
| − | [[Category:Number theory]]
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