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Euler's totient function has a lower bound for large values, but is there any way to pick out maximums for specific values of the function?

That is, how would I find the maximum number n such that phi(n) = 1000, for example?

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    You'd probably have to utilize the prime factorizations of the numbers $p-1$ for primes $p$.2011-10-31
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    The sequence you're looking for is at http://oeis.org/A006511 -- although there are no references there. There's some Mathematica code but it looks like it relies on Mathematica's "phiinv" function. You might want to try looking for references to the "inverse totient" function.2011-10-31
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    so that sequence suggests phi(7814) = 1000...is there no easy way to confirm that this is indeed the largest then?2011-11-01
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    You must be misreading the reference. Try to factor 7814, and see whether it's actually the case that $\phi(7814)=1000$.2011-11-01

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