The best time to observe a planet depends on a lot of factors. Important local circumstances include a free unclouded sky and absence of light pollution. The darkness of the background sky is important too.
Here we are going to ignore the above factors, but focus on another important factor: the apparent brightness of the planet.
The apparent brightness depends on these factors:
For inner planets, there is a trade off between the latter two factors, which both depend on the planet's elongation (i.e. the angle planet-Earth-Sun). So we may look for an elongation with optimum brightness.
Consider a planet orbiting the sun in a circular orbit with radius of . The question is:
At what elongation is the planet at its brightest?
If where , and are positive integers and is square free, submit as your answer.
Because a greater elongation offers the chance of observing against a darker background, the found value may serve as a lower bound. For Venus (at r = 0.72 AU) we find an elongation of 39.6° (at a phase of 26.7% - a crescent), but in practice larger elongations generally offer the best visibility. For Mercury ( at r = 0.38 AU) even more so: the largest elongation is the best.