MathDB

Is it possible to place

1995

different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime? I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it:
Suppose that we do have such a configuration

a_{1},a_{2},...a_{1995}

. WLOG,

a_{2}=p_{1}a_{1}

. Then

\frac{a_{2}}{a_{3}}= p_{2}, \frac{1}{p_{2}}

\frac{a_{3}}{a_{4}}= p_{3}, \frac{1}{p_{3}}

...

\frac{a_{1995}}{a_{1}}= p_{1995}, \frac{1}{p_{1995}}

Multiplying these all together,

\frac{a_{2}}{a_{1}}= \frac{\prod p_{k}}{\prod p_{j}}= p_{1}

Where

\prod p_{k}

is some product of the elements in a subset of

\{ p_{2},p_{3}, ...p_{1995}\}

. We clear denominators to get

p_{1}\prod p_{j}= \prod p_{k}

Now, by unique prime factorization, the set

\{ p_{j}\}\cup \{ p_{1}\}

is equal to the set

\{ p_{k}\}

. However, since there are a total of

1995

primes, this is impossible. We conclude that no such configuration exists.

Problem Statement