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February 10th, 2010, 03:39 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) 3uple totient ?(1,2,3)=13 (is prime) 4uple totient ?(1,2,3,1)=23 (is prime) 5ulpe totient ?(1,2,3,1,6)=47 (is prime) 6uple 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, 04:19 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, 05:40 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, 05:49 PM  #4  
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Re: An infinite number (?) of primes using the uple totient Quote:
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, 05:51 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, 06:04 PM  #6 
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  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? 
February 10th, 2010, 06:18 PM  #7  
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 Quote:
 
February 10th, 2010, 06:28 PM  #8  
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Re: An infinite number (?) of primes using the uple totient Quote:
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, 06:35 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, 06:38 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 

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infinite, number, primes, totient, uple 
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