- #1
- 19,546
- 10,287
This is wrong. Well, yes and no. Strictly speaking, this definition describes irreducibility. And irreducibility is different from primality. A prime number is actually a number, that if it divides a product, then it has to divide one of the factors. $$7\,|\,28=2\cdot 14 \;\Longrightarrow \;7\,|\,2\text{ or }7\,|\,14$$ $$4\,|\,12=2\cdot 6\text{ but }4\,\nmid \,2 \text{ and } 4\,\nmid \,6$$ It is a bit more complicated, but it is the correct definition. However, irreducible integers are prime integers and vice versa which is why the correct definition is replaced at school by the more handy one. However, there are domains in which this is not the case. The standard example is ##\mathbb{Z}[\sqrt{-5}]## where $$6=2\cdot 3=(1+\sqrt{-5})\cdot (1-\sqrt{-5})$$ is a decomposition into irreducible factors that are not prime.
I divided up parts of @fresh_42's latest Insight to facilitate discussion on eachBvU said:What is going on here ?
A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has no other factors besides 1 and itself.
To determine if a number is prime, you can check if it is only divisible by 1 and itself. One way to do this is by using a process called trial division, where you divide the number by every integer from 2 to the square root of the number. If none of these divisions result in a whole number, then the number is prime.
Yes, there are an infinite number of prime numbers. This was first proven by Euclid around 300 BC. The proof is based on the fact that if you multiply all the prime numbers together and add 1, the resulting number will either be prime or divisible by a new prime number that was not included in the original set.
Prime numbers are important in many areas of mathematics, including number theory, cryptography, and computer science. They also have practical applications in fields such as coding and data compression. Additionally, prime numbers play a crucial role in understanding the distribution of numbers and patterns in the natural world.
No, a number cannot be both prime and composite. A prime number, by definition, has only 2 factors (1 and itself), while a composite number has at least 3 factors. Therefore, a number cannot have both 2 and 3 factors at the same time.