[OT] Prime numbers

[Isaac M. Bavaresco]

Dear All,


Today I woke up with a silly idea about prime numbers in my head:

What is the proof that there are infinitely many prime numbers? One of
such proofs was in my mind.

Then I Googled and found Euclid's Proof. It is much similar to mine but
not exactly the same.


My proof:

Consider a finite list of consecutive prime numbers starting in 3: 3, 5,
7, 11, ..., Pn.

Let P be the product of all the prime numbers in the list.

Let Q = P + 2. Let's prove that Q is prime:

P + 1 is even (not prime)

P + 3 is multiple of 3 (not prime)

P + 5 is multiple of 5 (not prime)

...

P + Pn is multiple of Pn (not prime)


So Q cannot be multiple of any of the numbers in the list, thus Q is prime.


I found
(Wikipedia:<https://en.wikipedia.org/wiki/Largest_known_prime_number>)
that the largest known prime number is 2^74,207,281 − 1 which has
22,338,618 digits.

I found also a list of the first fifty million primes
<https://primes.utm.edu/lists/small/millions/>. If we multiply all the
numbers in this list, it will yield a number that has much more than 50
million digits and thus ought be much larger than the currently known
largest prime number.



Please help!!! Where is the catch? It cannot be that simple!


Cheers,

Isaac




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[smplx]

Post by Isaac M. Bavaresco
Dear All,
What is the proof that there are infinitely many prime numbers? One of
such proofs was in my mind.
Then I Googled and found Euclid's Proof. It is much similar to mine but
not exactly the same.
Consider a finite list of consecutive prime numbers starting in 3: 3, 5,
7, 11, ..., Pn.
Let P be the product of all the prime numbers in the list.
P + 1 is even (not prime)
P + 3 is multiple of 3 (not prime)
P + 5 is multiple of 5 (not prime)
...
P + Pn is multiple of Pn (not prime)
So Q cannot be multiple of any of the numbers in the list, thus Q is prime.
I found
(Wikipedia:<https://en.wikipedia.org/wiki/Largest_known_prime_number>)
that the largest known prime number is 2^74,207,281 − 1 which has
22,338,618 digits.
I found also a list of the first fifty million primes
<https://primes.utm.edu/lists/small/millions/>. If we multiply all the
numbers in this list, it will yield a number that has much more than 50
million digits and thus ought be much larger than the currently known
largest prime number.
Please help!!! Where is the catch? It cannot be that simple!

As I understand it your list of primes grows as you find new primes.
Problem is your solution is not finding all the primes between the last
prime in the list and the new prime you've just discovered. The next
prime you compute will not include these missing primes.

Regards
Sergio Masci

[Sean Breheny]

I am not 100% sure that your theorem is wrong, Isaac, but I think you are
making an incorrect assumption.

If I have two numbers A and B and I add them to produce C=A+B, and I then
have a number D, then:

if D divides A and D divides B THEN D divides C
However, just because D divides neither A nor B doesn't guarantee that it
doesn't divide C.
It IS true, though, that if D divides one and only one of A or B then it
does NOT divide C.

So, you cannot assume that the prime factors of Ptest=P1*P2*P3*P4*...Pn+2
are only from the set of {P1,P2,P3,...,Pn). There could be a prime factor
of Ptest which is larger than Pn.

Sean


On Wed, Sep 6, 2017 at 12:27 PM, Isaac M. Bavaresco <

Post by Isaac M. Bavaresco
Dear All,
What is the proof that there are infinitely many prime numbers? One of
such proofs was in my mind.
Then I Googled and found Euclid's Proof. It is much similar to mine but
not exactly the same.
Consider a finite list of consecutive prime numbers starting in 3: 3, 5,
7, 11, ..., Pn.
Let P be the product of all the prime numbers in the list.
P + 1 is even (not prime)
P + 3 is multiple of 3 (not prime)
P + 5 is multiple of 5 (not prime)
...
P + Pn is multiple of Pn (not prime)
So Q cannot be multiple of any of the numbers in the list, thus Q is prime.
I found
(Wikipedia:<https://en.wikipedia.org/wiki/Largest_known_prime_number>)
that the largest known prime number is 2^74,207,281 − 1 which has
22,338,618 digits.
I found also a list of the first fifty million primes
<https://primes.utm.edu/lists/small/millions/>. If we multiply all the
numbers in this list, it will yield a number that has much more than 50
million digits and thus ought be much larger than the currently known
largest prime number.
Please help!!! Where is the catch? It cannot be that simple!
Cheers,
Isaac
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[peter green]

Post by Isaac M. Bavaresco
Dear All,
What is the proof that there are infinitely many prime numbers? One of
such proofs was in my mind.
Then I Googled and found Euclid's Proof. It is much similar to mine but
not exactly the same.
Consider a finite list of consecutive prime numbers starting in 3: 3, 5,
7, 11, ..., Pn.
Let P be the product of all the prime numbers in the list.
P + 1 is even (not prime)
P + 3 is multiple of 3 (not prime)
P + 5 is multiple of 5 (not prime)
...
P + Pn is multiple of Pn (not prime)
So Q cannot be multiple of any of the numbers in the list,

True

Post by Isaac M. Bavaresco
thus Q is prime.

This is where you go wrong.

There may be prime numbers which are larger than Pn but smaller than Q. Some of these could conceivably be factors of Q.

For example

(3x5x7x11)+2 = 1157 = 13*89

So while this proves there is at least one prime that is not on your list (and hence proves there are infinitely many primes) it does not provide an efficient method for actually finding large primes.

[Sean Breheny]

Peter has a good point - if you also show that all integers have prime
factorizations (easy to do) then your method DOES prove that there are
infinitely many primes because it proves that there exists an integer Q,
larger than Pn, which is not divisible by any of the primes below (or equal
to) Pn - so either this number Q is itself prime, in which case you've
found a prime which is not in your original set, or if Q is not prime, it
must be divisible by at least one prime factor which is not contained in
your original set. Since Q is not even, this prime factor is not 2. Since
your set contains all primes between 2 and Pn, inclusive of Pn, you have
shown that there is at least one prime which is not in your set but is
larger than Pn.

Post by peter green

Post by Isaac M. Bavaresco
Dear All,
What is the proof that there are infinitely many prime numbers? One of
such proofs was in my mind.
Then I Googled and found Euclid's Proof. It is much similar to mine but
not exactly the same.
Consider a finite list of consecutive prime numbers starting in 3: 3, 5,
7, 11, ..., Pn.
Let P be the product of all the prime numbers in the list.
P + 1 is even (not prime)
P + 3 is multiple of 3 (not prime)
P + 5 is multiple of 5 (not prime)
...
P + Pn is multiple of Pn (not prime)
So Q cannot be multiple of any of the numbers in the list,

True

Post by Isaac M. Bavaresco
thus Q is prime.

This is where you go wrong.
There may be prime numbers which are larger than Pn but smaller than Q.
Some of these could conceivably be factors of Q.
For example
(3x5x7x11)+2 = 1157 = 13*89
So while this proves there is at least one prime that is not on your list
(and hence proves there are infinitely many primes) it does not provide an
efficient method for actually finding large primes.
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