## An Algorithm to generate primes ...???

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• ******* On Sat Nov 17, 2007 12:30 pm ((PST)) alby7e7 alby7e7@gmail.com alby7e7 Posted I suppose... if GCD(n! , n + 1) = 1 if GCD(n! , n + 1 ) != 1 also
Message 1 of 2 , Nov 18, 2007
*******

On Sat Nov 17, 2007 12:30 pm ((PST))

"alby7e7" alby7e7@... alby7e7

Posted

I suppose...
if GCD(n! , n + 1) = 1

if GCD(n! , n + 1 ) != 1

also (n+1)+1

so I do GCD(n!, n+2) = 1?
if yes I do the sum n! + n+2 also I do another GCD with n+3 and n!...

*******

Technically, this is a prime search, not a prime generation formula.
Another difficulty. GCD(n!,q) = 1 does not make sure that q is prime.

******

q might have a factor > n.

******

n! + n +1 = P

** but if n > 7

n + 1 > n! so I will do n+1+1+1+1+1+1+1+1+1...+1k

****

Apparently you meant to say

** but if n! < 7

n + 1 > n! so I will do n+1+1+1+1+1+1+1+1+1...+1k

******

while n+k > n! and GCD(n!,n+k ) = 1

n! + n + k = P

According to you is it true?

it can generate a lot of primes?

******

It's true that a lot of primes will be in this sequence,
but we won't know which ones are prime unless we do additional testing.

*****

*** for example : ***
EX.1
n=0

n! = 1 , n+1 = 2 GCD(1,1)= 1
1+1 = 2

EX.2

n=1

n!=1 n+1 = 2 GCD(1,2)= 1
1+2=3

EX.3

n=3

n! = 6
n+1 = 4 GCD(6,4)=2
GCD(6,4+1)=1
6+5 = 11

EX.4

n=4

n! = 24

n+1 = 5

GCD(24,5)=1

24+5 = 29

***

Example of when number in this sequence is not prime.

n = 12
n! = 479001600
n! + n + 1 = 479001613

479001613b = 29 * 2503 * 6599

Yet it is true that
GCD(479001600, 479001613) = 1

****
• ... I don t think that was the intended claim. I think the claim was probably meant to be that [GCD (n!, n+1 ) = 1] implies that [n! + n + 1] is prime. It
Message 2 of 2 , Nov 18, 2007
On Sun, 18 Nov 2007, Kermit Rose wrote:
> Another difficulty. GCD(n!,q) = 1 does not make sure that q is prime.

I don't think that was the intended claim. I think the claim was probably
meant to be that [GCD (n!, n+1 ) = 1] implies that [n! + n + 1] is prime.

It looks to me like maybe it's [finding/generating] series A073309,
http://www.research.att.com/~njas/sequences/A073309. (Primes of the form
n! + n + 1)

Although I'm not sure I can explain it clearly, it makes intuitive sense
to me for factors from 2 through n+1.

A.) n+1 is not a factor of n!, so it can't be a factor of [n! + (n+1)].
B.) Thinking of a number line, GCD=(n!,n+1) = 1 implies that the (n+1)
portion of [n! + (n + 1)] can't be evenly split into chunks which also fit
evenly into the (n!) part, so factors {2..n}, from n!, are not factors of
[n! + n + 1].

So I think factors from 2 through (n+1) can be ruled out. I'm not clear
about factors bigger than n+1 though. Maybe they can't be factors of [n!
+ n + 1], but I haven't noticed a reason why and I don't have more time to