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Infinite set of coprimes

by kureta
Tags: coprimes, infinite, solved
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kureta
#1
Feb2-07, 09:43 AM
P: 12
does any one know an infinite set of coprimes except for the elements of sylvester's sequence. S(n)=S(n-1)*(S(n-1)-1)+1, with s(0)=2
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Moo Of Doom
#2
Feb2-07, 11:03 AM
P: 367
The primes, maybe? :P
kureta
#3
Feb2-07, 11:06 AM
P: 12
yes. thanks but... well... nevermind

CRGreathouse
#4
Feb2-07, 07:16 PM
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Talking Infinite set of coprimes

The primes = 1 mod 4? The noncomposites? {2*3, 5*7, 11*13, 17*19, ...}?
kureta
#5
Feb2-07, 11:15 PM
P: 12
does any one know such a set AND its defining formula which gives us the nth term.
CRGreathouse
#6
Feb3-07, 04:22 PM
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Quote Quote by kureta View Post
does any one know such a set AND its defining formula which gives us the nth term.
The primes have many defining formulas, a fair number of which use only basic operations (say, addition, multiplication, factorials, and sine). To what end do you want this?
matt grime
#7
Feb5-07, 04:46 AM
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I don't have a reference to hand, but you can consider things like 2^n - 1. Indeed that is one proof that there are infintitely many primes.

Note 2^n - 1 = 2*(2^{n-1} -1) + 1, proving that they are coprime.
kureta
#8
Feb5-07, 05:18 AM
P: 12
2^2-1=3 and 2^4-1=15 and 15/3=5 so they are not all coprimes am i wrong?
matt grime
#9
Feb5-07, 12:42 PM
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Sorry, my mistake - consecutive terms are coprime, not every term. Duh. But there is something to do with things like this that demonstrates infintely many primes by producing coprimes. I don't have the reference to hand (I read it in 'Proofs from the Book' by Aigner and Ziegler).
ramsey2879
#10
Feb5-07, 09:20 PM
P: 894
Quote Quote by kureta View Post
does any one know such a set AND its defining formula which gives us the nth term.
a(n) = 5*2^(2n) +5*2^n + 1 is my guess. Prove or disprove
One thing that is certain, they can only be divisible by a prime ending in 1 or 9
ramsey2879
#11
Feb6-07, 08:43 AM
P: 894
Quote Quote by ramsey2879 View Post
a(n) = 5*2^(2n) +5*2^n + 1 is my guess. Prove or disprove
One thing that is certain, they can only be divisible by a prime ending in 1 or 9
Forget this too, If n = 8 2^n equals 9 and 2^2n = 4 mod 11.
5*(4+9) + 1 = 66. also there are other powers of 2 with the same residue mod 11 so those values for n give a(n) which are not coprime.
Maybe take n to be prime for the n in a(n) or something similar.
kureta
#12
Feb6-07, 09:50 AM
P: 12
ramsey2879 your reply was the kind that i was looking for. thanks. but taking n to be prime makes this formula useless for me. because my purpose is to find a set of coprimes generated by a simple function . and you should see "a different approach to primes" thread for the reason of my asking for such a set.


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