Analysis question -- Aren't all prime numbers not a product of primes?

[Clara Chung]

Homework Statement

I don't understand the lemma.

Homework Equations

The Attempt at a Solution

Isn't all prime number not a product of primes? The lemma doesn't make sense to me... Moreover, if m=2, m-1 is smaller than 2, the inequality also doesn't make sense. Please help me

 

[fresh_42]

Homework Statement

View attachment 212381

I don't understand the lemma.

Homework Equations

The Attempt at a Solution

Isn't all prime number not a product of primes? The lemma doesn't make sense to me... Moreover, if m=2, m-1 is smaller than 2, the inequality also doesn't make sense. Please help me

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]

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.

If prime number is not products of primes, how come the lemma is correct?...

 

[fresh_42]

Yes, $n=1\cdot p$ or $n=p$ count as product in its rigor meaning.

Formally it says that every integer $n\geq 2$ can be written as $n=\prod {p_i}^{m_i}$ with primes $p_i\; , \;m_i \in \mathbb{N}_0$ and $\sum m_i > 0$. This includes $n=p$, excludes $1$ and doesn't bother about any additional unnecessary factors $1$. The formulation in words is just this: saying in words what I wrote with symbols. Try to find another wording if you like. It will probably be longer. I would rather bother about the term Lemma because it's usual name is fundamental theorem of arithmetic - at least the existence part of it. (The uniqueness part is missing, and this requires to get rid of all eventual $1$'s.)