A new proof of Euclid's Theorem (2006)
fermatslibrary.com
A new proof of Euclid's Theorem (2006)
1–10 of 34 posts
Re: A new proof of Euclid's Theorem (2006)
#2Off-topic, but, what's the deal with all those dashes in the document?
Re: A new proof of Euclid's Theorem (2006)
#3Note: this paper is from 2006.
Re: A new proof of Euclid's Theorem (2006)
#4Old proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (some prime factor of (v_n + 1)).
"New" proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (v_n + 1).
Doesn't seem terribly different or more constructive or any such thing to me. I'd say both of these have the same fundamental content, just framed slightly differently.
[Just to head off what might superficially seem to be a significant distinction: Note that, in both proofs, we use the (constructive) fact that every integer > 1 has a prime factor, and even in the "new" proof, in order to actually extract an infinite stream of primes, one must actually carry out this process of finding prime factors for integers > 1 on demand.]
Re: A new proof of Euclid's Theorem (2006)
#5The idea of using `consecutive numbers are coprime` as the sole property for this concise proof is quite remarkable.
I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
Re: A new proof of Euclid's Theorem (2006)
#6The idea of using `consecutive numbers are coprime` as the sole property for this concise proof is quite remarkable. I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
What property does the classic proof use that the "new" proof does not?
Re: A new proof of Euclid's Theorem (2006)
#7Off-topic, but, what's the deal with all those dashes in the document?
Getting them too, using Firefox 43.0.4 on OS X Yosemite 10.10.3
Re: A new proof of Euclid's Theorem (2006)
#8Old proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (some prime factor of (v_n + 1)). "New" proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do thi…
Surprised this got published. This theorem is literally day-1 number theory. And unless I'm missing something, this paper is just rephrasing it in a slightly different way.
Also seems like a pretty baseless claim to say that this proof has never been done before.
Re: A new proof of Euclid's Theorem (2006)
#9The idea of using `consecutive numbers are coprime` as the sole property for this concise proof is quite remarkable. I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
What property does the classic proof use that the "new" proof does not?
[deleted]
Re: A new proof of Euclid's Theorem (2006)
#10[deleted]