From: veritasium
The Twin Prime Conjecture is a significant unsolved problem in mathematics [00:00:17]. It concerns a specific type of prime number pair.
Definition of Twin Primes
Twin primes are prime numbers that are separated by exactly one number [00:00:20]. Examples include 11 and 13, or 17 and 19 [00:00:20].
Properties and Rarity
As one progresses along the number line, prime numbers themselves occur less frequently, and twin primes are even rarer [00:00:27].
The Conjecture
The Twin Prime Conjecture proposes that there are infinitely many twin primes [00:00:34]. This means that, theoretically, you would never exhaust the supply of twin primes [00:00:38].
Current Status
As of the present, no one has successfully proven this conjecture to be either true or false [00:00:41]. It remains an unsolved question [00:25:00].
Connection to Undecidability
The Twin Prime Conjecture exemplifies a type of statement that may be true but cannot be proven [00:00:09]. This phenomenon is related to the concept of undecidability in mathematics [00:27:08].
- A Turing machine program could be conceptualized to attempt to solve the Twin Prime Conjecture [00:24:33]. Such a program would start with axioms and systematically construct all possible theorems [00:24:37]. If it generates the Twin Prime Conjecture, it would halt; otherwise, it would continue indefinitely [00:24:47].
- However, if the halting problem (the problem of determining if a program will ever halt) is undecidable, then solving the Twin Prime Conjecture this way is impossible [00:24:16], [00:26:53].
- Therefore, the Twin Prime Conjecture might be truly unsolvable, meaning mathematicians may never know if there are an infinite number of twin primes [00:27:08], [00:27:14]. This implies that there is no algorithm that can always determine whether a statement is derivable from the axioms [00:27:02].