What is a Conjecture?
In the field of mathematics, a conjecture is an assertion that is likely to be true but has not been formally proven. A conjecture is elevated to a theorem, if and only if it can be formally proven with the rules logic.
The Austrian philosopher Karl Popper, brought the word conjecture into the philosophy of science. In a philosophical context, a conjecture is a proposition that is thought to be true despite being primarily based on something inconclusive.
The word conjecture is also used colloquially to refer to a hunch or guess.
Christian Goldbach was a mathematician from Konigsberg who is most famous for the conjecture that he described in a letter to fellow mathematician Euler in 1742.
Goldbach proposed that "Every even integer greater than two can be expressed as the sum of two primes." For example, the number 4 is the sum of two prime numbers, 2 and 2. 6 can be expressed as the sum of 3 and 3. 8 = 5 + 3. 10 = 7 + 3 or 5 + 5.
This characteristic of even numbers appears true for any example you try, but it remains a conjecture precisely because it hasn't been formally proven. If you discover a counterexample, be sure to let us know!
Twin Prime Conjecture
A twin prime is a pair of prime numbers that have a difference of two. For example 3 and 5 are twin primes, as are 5 and 7, 11 and 13, 17 and 19, and so on. The Twin Prime Conjecture asserts that there are an infinite number of twin primes. Proving the twin prime conjecture has been notoriously difficult, and as of yet, no one has been able to successfully generate a proof.