The lands above 66° 33'N and 66° 33'S fall respectively within the Arctic and Antarctic circles. These latitudes define the approximate limit of the midnight sun and the polar night. In other words, for at least one Summer day, the Sun will not set for the full 24 hours, and for at least one winter day, the Sun will not rise for a full 24 hours.
As one travels towards the poles from the 66° 33 degree limit, the duration of this time without sunrise or sunset increases, until it can last for months. Indeed, at the poles themselves, the Sun only rises and sets once each year, with a half-year day and a half-year night.
There are today many Muslims living above the Arctic Circle, and they face the midnight sun/polar night phenomena for various amounts of time, ranging from one day up to a few months. For instance, the Russian city of Salekhard (pictured right) lies directly on the Arctic Circle and boasts a 20% Muslim population.
The mosque in Tromsø, Norway is the northernmost mosque in the world. It lies over 300 km inside the Arctic Circle and the Muslim community there experiences the midnight sun from 18 May to 26 July and perpetual night from 26 November to 15 January.
The Inuvik Mosque -- famous at holding the world record for the world's longest building move after traveling 4000 km from Winnipeg -- is approximately 200 km north of the Arctic Circle and the Muslim community there experiences an average of 56 days of continuous sunlight every summer and 30 days of polar night every winter.
Other important Muslim communities exist elsewhere above the Arctic Circle Canada, Alaska, Northern Europe, and Russia. The numbers of Muslims who live under these conditions is growing.
All Muslim scholars agree that whenever there is perpetual day or perpetual night for 24 hours or more, the prayer times during the affected days should be approximated. This is because the Prophet Muhammad (peace be upon him) said: "There will come a time when there will be a day like a year, a day like a month, and a day like a week…" The people asked him if during the day like a year, should they offer each prayer only once. He replied: "You should approximate the times." [Sahîh Muslim]
This means that the time between the prayers should be approximated so that all five prayers will be performed for each 24 hour period. This, however, is the extent of their agreement. The question remains as to how this estimation is to be achieved. Scholars are divided among three opinions:
 The prayer times should conform to the nearest populated locality where normal day and night are experienced. This is the opinion held by the majority of scholars.
Those who hold this view argue that since it is impossible to determine the prayer times by the position of the Sun in that particular locality, then looking to the closest place where there is normal sunrise and sunset gives the best approximation for how the prayer times should be.
 The day should be estimated according to a perfectly even day with 12 hours of daytime and twelve hours of nighttime. This is the view of a few scholars of the Hanbali school of thought.
These scholars argue that since it is impossible to determine the prayer times by the actual position of the Sun, the absolute mean should be taken, just as is the case with a number of other religious rulings requiring estimation. They cite, for example, the case with estimating the menstrual period for a woman who suffers from perpetual vaginal bleeding and never had a normal menstrual cycle to estimate from.
However, other scholars have pointed out that this is a weak comparison, since the closest inhabited place where there is day ad night brings a stronger resemblance to the locality in question than some hypotetical 12-hour day.
 The prayer times should be estimated so as to conform with the prayer times in Mecca for the same day. This opinion has been expressed by various scholars, as articulated in Fatâwâ al-Manâr (7/2578) and by al-Thunayân (p 24).
These scholars argue that Mecca has precedence of place. It is for Muslims "the mother of all cities" and the direction that Muslims turn to in their prayers. It is where Islam had its beginnings, and it should be what Muslims fall back on as a reference for prayer times in lands where normal day and night do not occur.
Other scholars have rejected this idea, arguing that there is no connection between Mecca's admittedly special qualities and prayer times in the Arctic. Mecca is far away from the latitudes where day and night fail to manifest, and therefore should not be used as a point of reference for estimating prayer times in those areas.
The Best Solution
It appears that the strongest of the opinions elaborated above is the first opinion: that of basing one's prayers upon the prayer times in the nearest populated locality where normal day and night are experienced. This is because the closest locale is the one that resembles conditions in the area the most. This is a rather compelling argument, while the supporters of the other two opinions provide very weak rationales in their defense.