Goldbach Conjectures
Goldbach Conjectures
[Six small panels with captions are arranged in an arch shape:]
[Caption under the arch:] Goldbach Conjectures
[Captions in the panels, from left to right:]
Extremely weak: Numbers just keep going
Very weak: Every number greater than 7 is the sum of two other numbers
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 strong: Every odd number is prime
Extremely strong: There are no numbers above 7

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.
Goldbach's weak and strong conjectures are a pair of real, unsolved problems relating to prime numbers (a number with exactly two positive divisors, 1 and itself). The comic states these under the labels "weak" and "strong".
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.
The title text gives the same treatment to the 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; 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.
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 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 tautology but more of a plain assertion. Randall has mentioned tautologies before in 703: Honor Societies .
Mathematician Christian Goldbach wrote a form of his conjecture (the "strong" one of the comic) in a letter to the famous Leonhard Euler in 1742. Euler replied that he considered it certainly true, but that he could not prove it.
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 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 10 30 , while all numbers below 10 30 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.
[Six small panels with captions are arranged in an arch shape:]
[Caption under the arch:] Goldbach Conjectures
[Captions in the panels, from left to right:]
Extremely weak: Numbers just keep going
Very weak: Every number greater than 7 is the sum of two other numbers
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 strong: Every odd number is prime
Extremely strong: There are no numbers above 7
