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 February 10th, 2010, 02:39 PM #1 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 An infinite number (?) of primes using the uple totient Hi, I discovered a way to generate (maybe) an infinite number of primes using the uple totient.(read this topic viewtopic.php?f=40&t=12181 ) Let the sequence U(n)={1,2,3,1,6,2,2,4,8,3,1,13,5...} If we begin by the 2 numbers of the sequence and then we add each time the number following we will obtain a sequence of primes : double totient ?(1,2)=5 (is prime) 3-uple totient ?(1,2,3)=13 (is prime) 4-uple totient ?(1,2,3,1)=23 (is prime) 5-ulpe totient ?(1,2,3,1,6)=47 (is prime) 6-uple totient ?(1,2,3,1,6,2)=61 (is prime) and so on ?(1,2,3,1,6,2,2)=79 ?(1,2,3,1,6,2,2,4)=107 ?(1,2,3,1,6,2,2,4,=163 ?(1,2,3,1,6,2,2,4,8,3)=199 ?(1,2,3,1,6,2,2,4,8,3,1)=233 ?(1,2,3,1,6,2,2,4,8,3,1,13)=373 ?(1,2,3,1,6,2,2,4,8,3,1,13,5)=457 I have built the first 12 numbers. Can someone continue the sequence. If we have a huge sequence maybe some pattern or property will be discovered. Thank you for any help.
 February 10th, 2010, 03:19 PM #2 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Re: An infinite number (?) of primes using the uple totient Here is the continuation of U(n) 1,2,3,1,6,2,2,4,8,3,1,13,5,3,29,14,7,3,29,13,3,13, 39,3,22,2,3,2.... The sequence of prime generated : 5 13 23 47 61 79 107 163 199 233 373 457 491 541 563 607 641 691 739 773 821 853 887 911 937 971 997 I think that the sequence of primes is as infinite as U(n).
 February 10th, 2010, 04:40 PM #3 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: An infinite number (?) of primes using the uple totient What is U(n)?
February 10th, 2010, 04:49 PM   #4
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Re: An infinite number (?) of primes using the uple totient

Quote:
 Originally Posted by CRGreathouse What is U(n)?
U(n) is a sequence created on the basis of an algo very simple

We begin by ?(1,x)
We replace x by a number starting each time by 1
So we find 2 ---> ?(1,2) is equal to 5 (prime number)
Then we continue ?(1,2,x)
We find 3 ---> ?(1,2,3) = 13 (prime)
and so on ....

Once we have a big sequence U(n) then we can try to explain why to find some pattern and so on

 February 10th, 2010, 04:51 PM #5 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Re: An infinite number (?) of primes using the uple totient Here is the last one : U(n)=1,2,3,1,6,2,2,4,8,3,1,13,5,3,29,14,7,3,29,13, 3,13,39,3,22,2,3,2,3,39,3,2,13,14,7.... If you look at it some numbers are repeated
February 10th, 2010, 05:04 PM   #6
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Re: An infinite number (?) of primes using the uple totient

Here is a diagramme of the prime generated :

[attachment=0:1e4wh9yc]primegnertotient.GIF[/attachment:1e4wh9yc]

What do you think about it?
Attached Images
 primegnertotient.GIF (6.8 KB, 374 views)

February 10th, 2010, 05:18 PM   #7
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Re: An infinite number (?) of primes using the uple totient

Quote:
 Originally Posted by momo We begin by ?(1,x) We replace x by a number starting each time by 1 So we find 2 ---> ?(1,2) is equal to 5 (prime number) Then we continue ?(1,2,x) We find 3 ---> ?(1,2,3) = 13 (prime) and so on ....
Why 3? ?(1, 2, 2) = 11 is prime.

February 10th, 2010, 05:28 PM   #8
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Re: An infinite number (?) of primes using the uple totient

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by momo We begin by ?(1,x) We replace x by a number starting each time by 1 So we find 2 ---> ?(1,2) is equal to 5 (prime number) Then we continue ?(1,2,x) We find 3 ---> ?(1,2,3) = 13 (prime) and so on ....
Why 3? ?(1, 2, 2) = 11 is prime.
You are right.
When you work with Excel and you enter your data manually it happens.
Sorry for the mistake.
I have to recompute all the numbers!!!

 February 10th, 2010, 05:35 PM #9 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Re: An infinite number (?) of primes using the uple totient Can you please send me the right sequence (minimal sequence because we can have a lot of sequences). Using Rapidshare will be very quick. Thank you!
 February 10th, 2010, 05:38 PM #10 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: An infinite number (?) of primes using the uple totient I don't do Rapidshare, but here are the first 100: 1, 2, 2, 2, 2, 2, 3, 8, 3, 22, 2, 7, 3, 8, 10, 1, 4, 6, 4, 2, 1, 1, 7, 2, 16, 4, 1, 15, 12, 2, 10, 5, 3, 11, 3, 3, 3, 5, 5, 22, 3, 13, 6, 11, 33, 27, 16, 2, 2, 9, 5, 10, 1, 21, 5, 22, 7, 20, 12, 8, 12, 3, 10, 4, 24, 3, 8, 1, 4, 14, 13, 22, 7, 31, 24, 16, 9, 7, 6, 14, 4, 2, 6, 3, 14, 2, 9, 5, 18, 74, 23, 24, 8, 16, 3, 24, 5, 2, 4, 7 This comes from the following Pari code: Code: phiset(v)=my(s=sum(i=1,#v,v[i]));sum(i=1,s,sum(j=1,#v,gcd(v[j],i)==1)) find(v)=v=vector(#v+1,i,if(i<=#v,v[i],1));for(i=1,9e9,v[#v]=i;if(isprime(phiset(v)),return(i))) v=[1];for(i=2,100,v=concat(v,find(v)));v You can run this to larger values if you like, just replace the "100" with something larger.

 Tags infinite, number, primes, totient, uple

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