If math is a such a definite subject with solid answers, how are there still unsolved math problems? How do people even come up with them?
In: Mathematics
A pattern you’ll notice with a lot of the examples given in this thread: often times the trouble is with **infinity**. If you ask for example, does the Fibonacci sequence contain any square numbers besides 144 (12×12)? I can write out the first couple numbers in the sequence, or have a computer generate the first billion – and each one is trivial to check if it’s a square – but it’s fundamentally impossible to check ALL of them, because the sequence is infinite.
The only way to solve such a thing is come up with a mathematical argument – a proof – that employs some clever logic to prove something about an infinite set. As a very simple example, consider the question, “are there any even prime numbers besides 2?”. We can answer this by saying, *suppose there were such a number. Then since it’s even, it can be divided by two – and since it can be divided by 2, it can’t be a prime!* So we have proven something about ALL numbers, even though we never had to check them individually. A slightly harder problem in this vein, *is there a biggest prime number?*
Problems like this arise all the time when mathematicians are just playing around – exploring patterns, asking questions, finding neat arguments that then lead to other natural questions. Some of the most famous unsolved problems are famous because, if we knew the answer, it would unlock truths about a lot of other related questions. (An example is the [“P vs NP” problem](https://en.wikipedia.org/wiki/P_versus_NP_problem) in computer science)
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