## Re: sqrt(Phi(n,b)) for n>2 and b>1

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• ... There was an error in Part (6), giving an upper bound for the probability of a solution with n 100, sorry. ... [here I should have computed:]
Message 1 of 55 , Dec 22, 2010
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> This is the best that I could do, by way of an heuristic argument
> that the probability of another (n,b) pair is less than 1/10^6.

There was an error in Part (6), giving an upper bound
for the probability of a solution with n > 100, sorry.

Here I correct it, with no change to the conclusion:

> 6) For n > 6 we know from
> http://mathworld.wolfram.com/TotientFunction.html
> that eulerphi(n) > sqrt(n).
> Hence I estimate the probability of a solution
> with n > 100 as less than

[here I should have computed:]

{print(precision(sum(n=10^2,10^4,1./4^(eulerphi(n)/2))
+suminf(n=10^4,1/2^(sqrt(n)/2)),1));}

0.0000000005135754773441919258

[with the same conclusion:]

> which is far less than the estimate 2/10^7, in Part (4).
>
> 7) In conclusion, I believe that there is a probability
> of less than 1/10^6 for finding a second integer pair (n,b),
> with n>2 and b>1, such that sqrt(Phi(n,b)) is an integer.

David
• We have written up (some of) our findings: http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752 Kevin and David
Message 55 of 55 , Feb 11, 2011
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We have written up (some of) our findings:

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752

Kevin and David
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