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Problem on euler's totient fumction

Source: 2009 József Wildt International Mathematical Competition

April 15, 2020

number theorytotient functionfunction

Problem Statement

Let

\Phi

denote the Euler totient function. Prove that for infinitely many

k

we have

\Phi (2^k+1) < 2^{k-1}

and that for infinitely many

m

one has

\Phi (2^m+1) > 2^{m-1}