Assalamualaikum ustaz. How should a person pray in a region where the day and night each could reach up to six months? This happens in the arctic region.
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions and all those who follow his footsteps until the Last Day.
We start with the statement of Allah SWT:
وَأَقِيمُوا الصَّلَاةَ وَآتُوا الزَّكَاةَ وَارْكَعُوا مَعَ الرَّاكِعِينَ
“And establish prayer and give zakah and bow with those who bow [in worship and obedience].” 
According to al-Imam Abu al-Husain al-Subki, this verse is the ijab evidence (obligates) for prayer for everyone, believers and even children, except that for them is ta’dib (an obligatory education on their parents) while for the obligation on the child it is yet to happen for as long as they aren’t mumayyiz, haven’t reached puberty and has a logical rational mind. 
And Allah SWT also stated:
إِنَّ الصَّلَاةَ كَانَتْ عَلَى الْمُؤْمِنِينَ كِتَابًا مَّوْقُوتًا
“Indeed, prayer has been decreed upon the believers a decree of specified times.” 
Both verses are evidence for the sanctioning of prayer on every Muslim in this world. Fulfilling prayer is one of the obligations and fardhu ‘ain on every mukallaf Muslim, that is any Muslim individual who has reached puberty and is of sound mind and pure. 
Allah SWT further states:
وَمَا أُمِرُوا إِلَّا لِيَعْبُدُوا اللَّهَ مُخْلِصِينَ لَهُ الدِّينَ حُنَفَاءَ وَيُقِيمُوا الصَّلَاةَ وَيُؤْتُوا الزَّكَاةَ ۚ وَذَٰلِكَ دِينُ الْقَيِّمَةِ
“And they were not commanded except to worship Allah, [being] sincere to Him in religion, inclining to truth, and to establish prayer and to give zakah. And that is the correct religion.” 
The word “hunafa” in the above verse refers to people who are istiqamah (consistent) in their religion, then Allah commanded them to worship Him sincerely as the condition when praying and giving zakah. 
Prayer is worship that cannot be left in any type of situation. And the time for them has been set by Allah SWT according to the above statement of Allah SWT.
The Prophet PBUH also explained comprehensively the times for the five obligatory prayers. From Abdullah bin Amru bin al-‘As, the Prophet PBUH said:
وَقْتُ الظُّهْرِ إِذَا زَالَتْ الشَّمْسُ وَكَانَ ظِلُّ الرَّجُلِ كَطُولِهِ مَا لَمْ يَحْضُرْ الْعَصْرُ وَوَقْتُ الْعَصْرِ مَا لَمْ تَصْفَرَّ الشَّمْسُ وَوَقْتُ صَلَاةِ الْمَغْرِبِ مَا لَمْ يَغْرُبْ الشَّفَقُ وَوَقْتُ صَلَاةِ الْعِشَاءِ إِلَى نِصْفِ اللَّيْلِ الْأَوْسَطِ وَوَقْتُ صَلَاةِ الصُّبْحِ مِنْ طُلُوعِ الْفَجْرِ مَا لَمْ تَطْلُعْ الشَّمْسُ فَإِذَا طَلَعَتْ الشَّمْسُ فَأَمْسِكْ عَنْ الصَّلَاةِ فَإِنَّهَا تَطْلُعُ بَيْنَ قَرْنَيْ شَيْطَانٍ
“The time of the noon prayer is when the sun passes the meridian and a man’s shadow is the same (length) as his height, (and it lasts) as long as the time for the afternoon prayer has not come; the time for the afternoon prayer is as long as the sun has not become pale; the time of the evening prayer is as long as the twilight has not ended; the time of the night prayer is up to the middle of the average night and the time of the morning prayer is from the appearance of dawn, as long as the sun has not risen; but when the sun rises, refrain from prayer for it rises between the horns of the devil.” 
In any situation, the prayer must be performed in its time. And the prayer performed before its time is invalid and must be repeated if there is still time for it remaining, but if the time has ended then he has to replace (qadha’) the prayer. 
For the question asked, the daytime in the arctic region could reach up to 6 months and one couldn’t differentiate when the sun rises and sets. Such a situation makes it hard to determine the time for prayer. Considering this situation, scholars provide two solutions so that prayer can be performed at their prescribed times.
First: The time for prayer can be determined following the nearest country with day and night that can be differentiated in 24 hours. This is according to the istinbat from the hadith narrated by Talhah bin Ubaidillah. The Prophet PBUH said:
خَمْسُ صَلَوَاتٍ فِى الْيَوْمِ وَاللَّيْلَةِ
“The five times of prayer every day and night.” 
This opinion is also based on the Isra’ and Mikraj incident where the Prophet PBUH was first gie=ven 50 times of prayer. The Prophet went back and forth until it is reduced to 5 times. Until Allah SWT states:
يَا مُحَمَّدُ إِنَّهُنَّ خَمْسُ صَلَوَاتٍ كُلَّ يَوْمٍ وَلَيْلَةٍ
“O Muhammad, there are five prayers every day and night.” 
Thus, any country nearest with its day and night can be differentiated in 24 hours, then follow the time for the prayer of that country in determining the time for prayer. For example, among the nearest countries is Sweden. In Sweden, the sunrise is at 3.51 in the morning and the sun sets at 9.40 p.m. Thus, if calculated the time for ZOhor prayer would be at 12.45 p.m., Asar at 5.18 p.m. and Isyak at 11.23 p.m.
Second: Following the time of prayers in Mecca. The reason is the time in Mecca is more muktadil when compared to other countries around the arctic. This is the place where the revelation is sent. This could be seen when Allah SWT declared Mecca as Ummul Qura the centre for its surrounding areas. Allah SWT states:
وَلِتُنذِرَ أُمَّ ٱلْقُرَىٰ وَمَنْ حَوْلَهَا ۚ
“That you may warn the Mother of Cities (Ummul Qura’) and those around it.” 
To conclude, we are inclined towards the second opinion because the times in Mecca is more muktadil compared to other countries around the arctic. It would also give wider facilitation compared to the time in the neighbouring countries of which the duration between each time of prayer is close and could be a burden.
However, we advise you to first refer to the Muslim authority bodies there (if there are any) as to not cause any misunderstanding and contradictions.