How Many Days Can You Shorten Prayers?
Question

During the summer vacation, we went on a trip for twenty days, and during this time we shortened our prayers because the Sahabah (may Allah be pleased with them) went on trips for six months and shortened their prayers. Is this view correct? Do we have to make up the days that have passed?
Praise be to Allah.Qasr Prayer: For how many days?
The majority of scholars are of the view that a traveller may avail himself of the concessions of travel so long as he is not staying in the place to which he has gone for four or more days , whether he travels for work, medical treatment, a vacation or any other reason. 
Ibn Qudamah (may Allah have mercy on him) said: The well-known view of Ahmad (may Allah have mercy on him) is that the period during which the traveller must perform his prayers in full with the intention of staying is a period in which there are more than twenty-one prayers. And it was narrated from him that if he intends to stay for more than four days he must perform his prayers in full, but if he intends to stay less than that, he may shorten them . This is also the view of Malik and al-Shafi’i.” (Al-Mughni, 2/65) 
It says in Fatawa al-Lajnah al-Daimah (8/109): 
“The basic principle is that the traveller who is actually travelling is the one who is granted a concession allowing him to shorten the four-rak’ah prayers , because Allah says (interpretation of the meaning): 
“And when you (Muslims) travel in the land, there is no sin on you if you shorten As-Salah (the prayer)” [al-Nisa 4:101]
And Ya’la ibn Umayyah said: I said to ‘Umar ibn al-Khattab (may Allah be pleased with him): “ ‘And when you (Muslims) travel in the land, there is no sin on you if you shorten the prayer if you fear that the disbelievers may put you in trial (attack you),’” He said: I wondered the same thing as you, and I asked the Messenger of Allah (peace and blessings of Allah be upon him) about it, and he said: “It is a charity that Allah has bestowed upon you, so accept His charity.” (Narrated by Muslim)
The one who stays in a place for four days and nights or less also comes under the ruling of the traveller who is actually travelling, because it is proven in the hadith of Jabir and Ibn 'Abbas (may Allah be pleased with them) that the Prophet (peace and blessings of Allah be upon him) came to Makkah on the fourth of Dhu’l-Hijjah for the Farewell Pilgrimage, and he (peace and blessings of Allah be upon him) stayed for the fourth, fifth, sixth and seventh, and he prayed Fajr in al-Abtah on the eighth day, and he shortened his prayers during those days. He had formed the intention to stay, as is well known. So everyone who travels and intends to stay for this length of time that the Prophet (peace and blessings of Allah be upon him) stayed, or less, may shorten his prayers. Whoever intends to stay longer than that should offer the prayers in full, because he does not come under the heading of a traveler. 
But if a person stays for longer than four days during his trip and has not formed the intention to stay, rather he has resolved that when he has finished his business he will go back, like one who stays in a place to engage in jihad against the enemy, or has been detained by the authorities or by sickness, but he has resolved that when the jihad ends with victory or a peace treaty, or the sickness or enemy power that is keeping him there ends, or he sells all his goods and so on, he will go back, then he is regarded as a traveller, and he may shorten the four-rak’ah prayers, even if that period is long, because it is proven that the Prophet (peace and blessings of Allah be upon him) stayed in Makkah during the year of the Conquest for nineteen days, during which he shortened his prayers, and he stayed in Tabuk for twenty days to fight the Christians, and he led his Companions in shortened prayers, because he had not resolved to stay, rather his intention was to leave once he had finished his business.”  
Did the Companions travel for vacations?
The Sahabah (may Allah be pleased with them) did not travel for a vacation for six months as mentioned in the question. Rather they would travel for jihad for the sake of Allah, or to seek knowledge, or to seek a halal provision, and other religious and worldly interests, such as Ibn ‘Umar (may Allah be pleased with him) who stayed in Azerbaijan for six months, and snow kept him from entering, and he shortened his prayers. 
Should you make up the prayers that you shortened?
You do not have to make up the prayers that you shortened during this time, because it is possible that you may come under the category of travelling. The Standing Committee was asked about a person who was sent by Saudi to Germany, who stayed there for nearly one and half years and he shortened his prayers. They replied: 
“You do not have to make up the prayers that you shortened or delayed or joined with other prayers, because it is possible that you may come under the heading of travelling. But in the future, you should pray the four-rak'ah prayers in full and offer every prayer on time, because the ruling of travel no longer applies to you, because you have resolved to stay, and you have resolved to stay for more than four days. So you have to pray in congregation if possible, and do not pray alone.” (Fatawa al-Lajnah al-Daimah, 8/155) 
For more, please see this category: Prayer of the Traveller
And Allah knows best.