The Lemma states that all integers are a product of primes. Of course you have to rule out the units ##\pm 1## as one can always add arbitrary many of them. This would make no sense. Similar it isn't important to treat positive and negative numbers separately, so the condition ##\ge 2## makes sense. I do not understand what you mean by "all prime number a product of primes". It is, if you consider ##p=p## as a product, but here proper products without units are meant.Clara Chung said:
If prime number is not products of primes, how come the lemma is correct?...fresh_42 said:
A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has only two factors: 1 and the number itself.
No, by definition, a prime number cannot be a product of other prime numbers. It can only be divided by 1 and itself, so it cannot be written as a product of other numbers.
This is a fundamental theorem in number theory, called the Fundamental Theorem of Arithmetic. It states that every positive integer can be expressed as a unique product of primes. Therefore, prime numbers cannot be written as a product of primes because they are already prime themselves.
No, there are no exceptions. All prime numbers follow this rule and cannot be written as a product of other primes.
This concept is important because it helps us understand the fundamental building blocks of numbers. It also has many applications in cryptography, where prime numbers are used to ensure the security of codes and messages.