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• ## Re: [PrimeNumbers] Prime-related conjectures

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• The conjectures are false, to see it take the prime 653 = p_119 , we see that: p-1 = 4 .163 p+1 = 6 . 109 but for all the primes p_i (i=1,...,158), we
Message 1 of 13 , Jun 24, 2002
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The conjectures are false, to see it take the prime 653 = p_119 , we see
that:
p-1 = 4 .163 p+1 = 6 . 109
but for all the primes p_i (i=1,...,158), we have that neither p_i-1 nor
p_i+1 are
divisible by 109 or by 163, thus
product( i=1, 119, p_i+a_i) is allways divisible by 109 or by 163 but
never by 109^2
or 163^2, and as a consequence it can not be a perfect square or a perfect
power.

to find other examples it is enough to look for primes q_1 such that both
q_2 = (3q_1-1)/2 and p = 6q_1-1 are prime but all the
numbers
2q_1 +- 1 or 2q_2 +- 1 are composite, and it is natural to conjecture that
there
are infinitly many.

Esteban Crespi de Valldaura

jbrennen escribiÃ³:

> Okay, here are some conjectures I just came up with. These might be
> original; I've never seen similar conjectures myself. Perhaps these
> can be proven -- I have some ideas, but haven't yet figured out if
> they amount to a rigorous proof. :-)
>
> Conjecture:
>
> Given a number n, there exists a sequence a_1 ... a_n consisting
> entirely of +1 or -1 values such that:
>
> product(i=1..n,p_i+a_i) is an integer square.
>
> In this case, p_i is the i'th prime: p_1=2, p_2=3, ...
>
> For instance:
>
> (2-1)*(3-1)*(5-1)*(7-1)*(11-1)*(13+1)*(17-1)*(19+1)*(23+1)*
> (29+1)*(31+1)*(37-1)*(41-1)*(43+1)*(47+1)*(53+1)*(59+1)*
> (61-1)*(67-1)*(71+1)*(73-1)*(79+1)*(83+1)*(89+1)*(97-1)
>
> is equal to 762821927239680000^2.
>
> Another conjecture:
>
> Same statement, but with integer cubes, and with the singular
> exception of n=2, for which no solution exists. For instance:
>
> (2+1)*(3+1)*(5+1)*(7+1)*(11-1)*(13+1)*(17+1)*(19-1)*(23+1)*
> (29+1)*(31+1)*(37-1)*(41+1)*(43+1)*(47+1)*(53+1)*(59+1)*
> (61-1)*(67-1)*(71-1)*(73-1)*(79+1)*(83+1)*(89-1)*(97+1)
>
> is equal to 1931334451200^3.
>
> Finally, for the general case (and frankly, I don't believe this one
> to even be true):
>
> For any integer X, there exists a threshold N such that if n>=N, there
> exists a sequence a_1 ... a_n consisting entirely of +1 or -1 values
> such that:
>
> product(i=1..n,p_i+a_i) is an integer X-th power.
>
> Any thoughts?????
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

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