### Abstract:

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental
role in cosmology. The direct search of the exact solutions of the Einstein equation even in
this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of
the Einstein equation using a known solutions of some other equations which are equivalent
or related to the Einstein equation. In this work, we establish the relationship the Einstein
equation with two other famous equations namely the Ramanujan equation and the Chazy
equation. Both these two equations play an imporatant role in the number theory. Using the
known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions
of the Einstein equation. The relation between the Friedmann equation and the equations of
the trefoil knot and figure-eight knot is established.

### Description:

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental
role in cosmology. The direct search of the exact solutions of the Einstein equation even in
this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of
the Einstein equation using a known solutions of some other equations which are equivalent
or related to the Einstein equation. In this work, we establish the relationship the Einstein
equation with two other famous equations namely the Ramanujan equation and the Chazy
equation. Both these two equations play an imporatant role in the number theory. Using the
known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions
of the Einstein equation. The relation between the Friedmann equation and the equations of
the trefoil knot and figure-eight knot is established.