Lemma's needed to prove the fundamental theorem of arithmetic

Question

Lemma's needed to prove the fundamental theorem of arithmetic

I'm reading the proof of the fundamental theorem of arithmetic in my textbook.

The proof is a proof by contradiction and is easy to follow, but they say that there are two lemma's (Lemma 3.4 and Lemma 3.5) needed to proof the fundamental theorem of arithmetic.

I do understand all the proofs, but I do not understand in which step of the proof of the fundamental theorem of arithmetic we need lemma 3.4 and lemma 3.5.

Can someone explain that to me?

Screenshot of the proof in my tekstbook:

As you can see, the lemma's are not used in the proof you have shown. So your question is the "empty question". Of course, if you go to the next page (uniqueness), then you will need them. — Dietrich Burde, Mar 22 '18 at 15:48
You see, now your question is explained in detail in the text. — Dietrich Burde, Mar 22 '18 at 15:51
@DietrichBurde Can you explain me why we need n to be the SMALLEST integer that cannot be written as the product of primes? — WinstonCherf, Mar 22 '18 at 15:54
Because this shows the existence of a factorization in the first part. We come to the desired contradiction this way. — Dietrich Burde, Mar 22 '18 at 15:57
Euclid's Lemma (your 3.5) is used in a common proof of uniqueness of prime factorizations, e.g. see here in the dupe. See also here — Bill Dubuque, Apr 16 '24 at 16:07

score 1 · Answer 1 · answered Mar 22 '18 at 16:10

Lemma 3.4 is used to prove lemma 3.5. Lemma 3.5 is used to prove the uniqueness of the Fundamental Theorem of Arithmetic. There $$\prod_{r=1}^u p_{i_r}=\prod_{a=1}^v q_{j_a}$$. Here , say $p_{i_k},1\leq k\leq u$. Then, $p_{i_k}|\prod_{a=1}^v q_{j_a}\Rightarrow p_{i_k}|q_{j_l}$, for some $l, 1\leq l \leq v$, becuse of lemma 3.5. Sincce both are primes then they must be equal.

Your query about the smallest $n$. We choose smallest because then in the $n=ab$, part where $1<a,b<n$, we can say confidently that both a and b both foolow fundamental theorem of arithmetic and prevent from considering any $n$.

Lemma's needed to prove the fundamental theorem of arithmetic

1 Answers1