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Question

Show that if k > 1 is an integer, then the equation τ(n)=k\tau(n)=k has infinitely many solutions.

Solution

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\hspace*{5mm}At first for any integer k>1k>1, there are infinitely many integers for which

τ(n)=k\tau(n)=k

\hspace*{5mm}Now let

n=pk1n=p^{k-1}

\hspace*{5mm}Then if mm is any positive integer with prime factorization

m=p1a1p2a2psam=p_{1}^{a_{1}} p_{2}^{a_{2}} \ldots p_{s}^{a}

\hspace*{5mm}Number of divisors of mm is

τ(m)=j=1s(aj+1)\tau(m)=\prod_{j=1}^{s}\left(a_{j}+1\right)

\hspace*{5mm} τ(n)=k\tau(n)=k since there are infinitely many primes therefore there are infinitely many nn such that

n=pk1 and τ(n)=kn=p^{k-1} \text{ and } \tau(n)=k

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