Editing 1310: Goldbach Conjectures
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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. | 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. | ||
β | It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime | + | It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; for example, 23 is prime, but 25 is not. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first". |
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]]. | 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]]. |