Talk:Fundamental theorem of arithmetic

message: First, one must consider what numbers actually are interesting: 1 is unity, 2 is the smallest even and smallest prime number, 3 is the smallest odd prime number, 4 is the first whole number that is the square of another integer, 5 is significant in Biology (many vertebrates have some version or form of 5 digits), 6 is the first perfect number, 7 is a historically important number in many religions and mythologies, etc, etc....
Assume that one can eventually find and group all of the Uninteresting numbers. By the principle of Well Ordering, they can be arranged from smallest to largest. Let's look carefully at this group:
The first number that one comes across is the smallest of all the Uninteresting numbers known to exist in the universe. Well, that's interesting....
--Daydreamer302000 16:37, 28 February 2007 (UTC)
replies: text: We have an article about the interesting number paradox. PrimeHunter (talk) 10:34, 22 October 2018 (UTC)
replies: text: But then the next Uninteresting number is really the smallest, which robs the previous one of all interest.  BMJ-pdx (talk) 14:57, 26 July 2021 (UTC)
thread_title:  Joke: There are no Uninteresting Numbers 

message: The proof shows that any integer s has a unique factorization of primes.
At some point the proof introduces an integer t  and says: 
["t= (q1-p1)(q2....qn)
and note that 1 < q2 <= t < s. Therefore t must have a unique factorization..."]
(by the way , it was also mentioned that q1>p1)
(1) Given that we are trying to prove unique factorization and have not proved it at 
this point in the proof, 
why can we say at this stage that t has a *unique* factorization?
(2) At this point, its easy to see that t has a *different* factorization than s.
(3) But It seems hard to understand that it has a *unique* factorization. 
(4) After all the whole point of the proof is to prove that s has a 
unique factorization.
(5) At that point of the proof (where t=(q1-p1)(q2....qn)) is introduced, s hasnt yet been proven to have a unique factorization.
Is it safe to say at that time then that t has a unique factorization?
Looking again at t=(q1-p1)(q2....qn),
what could (q1-p1) be at this point?
It could be a composite number, then we are left with:
t = [some set of primes] (different from s) and how do we know that t is uniquely represented by those set of primes, 
given that we have not proved this in general yet?
Can anyone help with this?
Thanks!
Mooingzebra (talk) 20:03, 19 December 2018 (UTC)
replies: text: It has been supposed that s is the smallest integer that does not have a unique factorization. Thus, as t < s, it has a unique factorization. D.Lazard (talk) 20:36, 19 December 2018 (UTC)
thread_title:  Question about "Elementary proof of uniqueness"  

message: :The following is copied from my TP:
:I regret if my edit summary was perceived as offensive. Of course I agree to the intended meaning being straightforward, but nevertheless I think the formulation is not sufficiently rigorous to remain in a WP-article. Furthermore, I agree to the opinion, stated in this discussion that  requires more machinery than the current formulation. Purgy (talk) 11:25, 15 February 2019 (UTC)
replies: text:  I don't know what "not sufficiently rigorous" could possibly mean in this context; the statement (with the missing words "of primes" added) is completely correct and rigorous.  However, I agree that there doesn't seem any compelling reason to add the sentence.  --JBL (talk) 16:01, 15 February 2019 (UTC)
thread_title:  Use of "multisets" 

message: i have a theorem in my textbook which has been derived from the fundamental theorem of Arithmetic. the theorem is as follows" let p be a prime number and a be a positive number integer. if p divides a², then p divides a." which theorem is this?Huzaifa abedeen (talk) 05:33, 30 September 2020 (UTC)
replies: text: It is a special case of Euclid's lemma. D.Lazard (talk) 08:41, 30 September 2020 (UTC)
thread_title:  which theorem is this? 

message: In the proof of uniqueness is the following phrase "We see p1 divides q1 q2 ... qk, so p1 divides some qi by Euclid's Lemma".  However the article on Euclid's Lemma linked to only talks about the case where a prime divides the product of two numbers.  While it's not difficult to extend the proof to multiple numbers this should be clearer. — Preceding unsigned comment added by Ximera~enwiki (talk • contribs) 10:38, 13 April 2021 (UTC)
thread_title:  Unclear reference to Euclid's Lemma 

message: Isn't there a name for the set comprising a number's prime factors? I see nothing in the various prime/composite-related articles.  BMJ-pdx (talk) 15:02, 26 July 2021 (UTC)
replies: text: , see an answer to the same question in Talk:Greatest common divisor D.Lazard (talk) 15:17, 26 July 2021 (UTC)
thread_title:  Name for a number's set of prime factors? 

message: It is written
:the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes
This is wrong. For example,
:(1+i)(2+i) = (1-i)(2i-1).
It is true only if one chooses one representative for each equivalence class of primes modulo the units. Or, one should add "(except for order and factors of units)", or similar. — MFH:Talk 13:04, 7 April 2023 (UTC)
replies: text: . D.Lazard (talk) 15:55, 7 April 2023 (UTC)
thread_title:  Incorrect statement in generalization to gaussian integers 

message: The History section states 
"While Euclid took the first step on the way to the existence of prime factorization, Kamāl al-Dīn al-Fārisī took the final step and stated for the first time the fundamental theorem of arithmetic."
In this sentence, two sources are cited:
8. A. Goksel Agargun and E. Mehmet Özkan."A Historical survey of the Fundamental Theorem of Arithmetic" (PDF). Historia Mathematica: 209. "One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first proposition."
9. Rashed, Roshdi (2002-09-11). Encyclopedia of the History of Arabic Science?. Routledge. p. 385. ISBN 9781134977246. "The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first time the fundamental theorem of arithmetic."
The first source however links only to pages 162-173 of "A Historical survey of the Fundamental Theorem of Arithmetic" and on page 172 the authors disagree that al-Farisi stated or proved the uniqueness of prime factorisation:
"In our estimation, al-Farisi neither stated nor proved uniqueness, and it was not his intention to do so."
So the first source disagrees with the second that al-Farisi stated and proved (the entirety of) the FTA. In fact, the authors directly cite and criticise a different work by Rashed, who provided the claim that al-Farisi stated and proved the FTA in the second source. Muisnotforyou (talk) 16:10, 1 February 2024 (UTC)
thread_title:  Contradicting/incomplete sources in History section