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-   -   Euclid's proof of the infinite number of primes (https://www.mersenneforum.org/showthread.php?t=6090)

 troels munkner 2006-07-10 14:34

Euclid's proof of the infinite number of primes

How to understand Euclid's proof of the infinite number of primes.

 drew 2006-07-10 14:49

[QUOTE=troels munkner]How to understand Euclid's proof of the infinite number of primes.[/QUOTE]
Euclid's proof is very straightforward.

Let's say there is a finite number of primes. (2, 3 and 5, for example). 2n+1 is never divisible by 2. 3n+1 is never divisible by 3. 5n+1 is never divisible by 5.

Now, from the above, 2*3*5+1 is not divisible by 2, 3 or 5, so it must either be:

a. Prime
b. Divisible by other prime factors which are not 2, 3 or 5

In either case, there are more primes than simply 2, 3 and 5.

Which means that whenever you have a finite number of primes, you can find 1 more and repeat the process.

Drew

 troels munkner 2006-07-11 08:32

I try again to submit the (new) thread.

All the best,

troels

 troels munkner 2006-07-11 08:51

Euclid's proof

1 Attachment(s)
Unfortunally the three attachments were missing.
I will try to submit them in separate threads.

All the best,

troels

 troels munkner 2006-07-11 09:00

Euclid's proof (II)

1 Attachment(s)
The next attachment

 troels munkner 2006-07-11 09:36

Euclid (III)

1 Attachment(s)
The final attachment,

All the best,

troels

 Jens K Andersen 2006-07-12 00:52

How fortunate I already understood the proof. Otherwise I would be very confused now.

The 3 zip files are Word documents. The last 2 are diagrams. A quote from the 1st:
"Euclid (and most other mathematicians) have assumed that 2 and 3 are primes.
But I claim, that 2 and 3 are not possible primes and should not be considered as
“primes”."

In the [URL="http://mersenneforum.org/showpost.php?p=81325&postcount=2"]words of Paul[/URL]: Humpty-Dumpty alert!

 Patrick123 2006-07-12 08:12

[QUOTE](6*m +1) [one third of all integers][/QUOTE]:no:

Definitely a Humpty-Dumpty alert is required.

 troels munkner 2006-07-13 12:20

[QUOTE=Jens K Andersen]How fortunate I already understood the proof. Otherwise I would be very confused now.

The 3 zip files are Word documents. The last 2 are diagrams. A quote from the 1st:
"Euclid (and most other mathematicians) have assumed that 2 and 3 are primes.
But I claim, that 2 and 3 are not possible primes and should not be considered as
“primes”."

In the [URL="http://mersenneforum.org/showpost.php?p=81325&postcount=2"]words of Paul[/URL]: Humpty-Dumpty alert![/QUOTE]

You have better read the original publication, - and be more polite.

troels munkner

 troels munkner 2006-07-13 12:26

[QUOTE=drew]Euclid's proof is very straightforward.

Let's say there is a finite number of primes. (2, 3 and 5, for example). 2n+1 is never divisible by 2. 3n+1 is never divisible by 3. 5n+1 is never divisible by 5.

Now, from the above, 2*3*5+1 is not divisible by 2, 3 or 5, so it must either be:

a. Prime
b. Divisible by other prime factors which are not 2, 3 or 5

In either case, there are more primes than simply 2, 3 and 5.

Which means that whenever you have a finite number of primes, you can find 1 more and repeat the process.

Drew[/QUOTE]

I know of course Euclid's "proof". But I went behind the statement and
studied it in more details. Please, look up the attachments which were not
If you can read the attachments, you will see a new view of integers.

Y.s.

troels

 R.D. Silverman 2006-07-13 14:30

[QUOTE=troels munkner]I know of course Euclid's "proof". But I went behind the statement and
studied it in more details. Please, look up the attachments which were not