Editing 1310: Goldbach Conjectures
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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. | 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. | ||
β | 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 ( | + | 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 (numbers with exactly two divisors, 1 and itself). The comic states these under the labels "weak" and "strong". |
* 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. | * 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. | ||
* 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). | * 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). |