Basic Examples (2) 

Find all integers m such that ϕ(m)=1000:

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Check:

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Scope (2) 

Find all 6535 integers m such that ϕ(m)=10²⁰:

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Verify that these are the correct solutions:

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It takes only a fraction of second to find all solution for the numbers below on a modern computer:

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Properties and Relations (2) 

EulerPhiInverse returns an empty list for all odd integer except 1:

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List non-totients:

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Show that non-totients have no preimage under Euler's totient function:

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Possible Issues (2) 

EulerPhiInverse only handles positive integers. Otherwise, it returns unevaluated:

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The number of solutions may grow rapidly for those inputs with prime factors p such that p-1 is highly composite:

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Increase the second argument to get all solutions for x such that EulerPhi[x]=10²⁸:

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Neat Examples (3) 

Numbers k such that EulerPhi[x]=k has exactly 2 solutions (OEIS A007366):

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Visualize the data:

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A pair of dual graph: two integers m and n are connected if ϕ(m)=n on the left and ϕ^-1(m)=n on the right:

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The graph on the left hand side seems denser than the right because the inverse of ϕ is a multivalued function:

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Number of integers k such that EulerPhi[k]=n (OEIS A072074):

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Plot the data and its logarithm approximation:

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