Why do we need to boil water for longer at higher altitudes to sterilize it? 
Boiling can be used as a pathogen reduction method that should kill all pathogens. Water should be brought to a rolling boil for $1$ minute. At altitudes greater than $6,562$ feet (greater than $2000$ meters), you should boil water for $3$ minutes.

I found this online, I am not sure of its validity. I just wanted to know if anyone had an idea why we need to boil water for longer at higher altitudes. I know it reaches boiling point quicker, but not why we need to boil it for longer. 
 A: Both existing answers correctly say that water boils at a lower temperature at high altitude. However they then both incorrectly say that boiling for longer will reach a higher temperature. It will not, unless you take action to increase the pressure. Boiling for a longer time merely maintains the temperature at the boiling point, whatever that may be. This would still be useful to kill bacteria etc, if there are some hardy ones than can survive at high temperature but not for long periods. 
It is possible to raise the boiling temperature by raising the pressure. However, although in the first version of my answer I put forward some homespun ideas on how to do that, I stand corrected (on grounds of safety) and now would not advise anyone to try it. Rather, be informed about the medical science as well as the physics (see answer by Ben Crowell).
The main thing to note is that boiling point increases with ambient pressure.
A quick look at data just now gave me that the air pressure falls from 100 kPa (sea level) to 70 kPa at 3000 metres height. At this pressure water will boil at about 90 degrees celcius. On top of the Alps (Mont Blanc) I estimate 85 degrees celcius. The boiling temperature and vapour pressure info is at 
https://en.wikipedia.org/wiki/Vapour_pressure_of_water
and I used an air pressure vs. height calculator at 
https://www.mide.com/pages/air-pressure-at-altitude-calculator
A: The basic science involved is this:


*

*The boiling point is depressed with lower pressure.

*You can't raise the temperature past the boiling point. Once the water starts to boil, the temperature sticks there.

*The ability to kill disease-causing microorganisms depends on both the temperature and the length of time for which the temperature is applied. You can trade one for the other.
However: --

Water should be brought to a rolling boil for 1 minute. At altitudes greater than 6,562 feet (greater than 2000 meters), you should boil water for 3 minutes.

The quoted claim is simply wrong. Folklore about boiling has become more and more exaggerated over time, like the folklore about how many words Eskimos have for snow. A temperature of 72 C is sufficient to kill all bacteria within 15 seconds [USDA 2004]. Some spores can remain viable when heated to higher temperatures, but the spores that can survive these conditions are not types that are capable of making you sick when you ingest them. Even at the elevation of Everest Base Camp, the boiling point of water is about 80 C, which is plenty high enough for this purpose.
People cooking at high elevations in the wilderness generally don't carry a thermometer with them, so they have no way to know when their water is at a temperature such as 72 C. That makes it easier to wait until it starts to boil vigorously, because then you know it's plenty hot enough. The time taken to get from 72 C to the boiling point is longer than the time it takes to kill all the bad bugs.
As a practical matter, the reduction of the boiling point is pretty minor at the elevations most of us visit (say 3000-5000 meters), and when people do go to much higher elevations, like 6000 meters or more, they need to conserve fuel, which they need in order to melt snow to make drinking water. They can't afford to boil every milliliter of water they're going to drink.
There is also the unspoken assumption that it's necessary to treat backcountry water in order to avoid getting sick. There is typically no need to purify water collected from natural sources in the wilderness. For example, in a survey of 69 sites in the U.S Sierra Nevada, every site had concentrations of Giardia cysts much too low to make anyone sick.[Rockwell 2002] The perception that backcountry water is unsafe to drink without treatment is folk wisdom that is controverted by the available scientific evidence, at least in the backcountry areas in the US such as the Sierra that have been extensively studied. Of course none of this applies outside of the backcountry, e.g., you certainly don't want to drink water in Arkansas that comes from runoff from a poultry farm.
When people do actually contract backpacker's diarrhea from exposure during a hiking trip, by far the most common reason is hand-to-mouth contamination.[Welch 1995] The most effective disease prevention measures are to wash your hands after pooping, refrain from sharing pots and pans, and use freezer-bag cooking so that food never goes in your pots.
References
Rockwell 2002 - Robert L. Rockwell, Sierra Nature Notes, Volume 2, January 2002,
USDA 2004 - National Advisory Committee on Microbiological Criteria for Foods: Requisite Scientific Parameters for Establishing the Equivalence of Alternative Methods of Pasteurization, USDA, 2004.
Welch 1995 - Thomas R. Welch and Timothy P. Welch, "Giardiasis as a threat to backpackers in the United States: a survey of state health departments," Wilderness and Environmental Medicine, 6 (1995) 162.
A: Boiling water works as a pathogen reduction method because the temperature destroys the essential structures, so to speak, of many bacteria. Since water boils at lower temperatures at higher altitudes, it is advised to let it boil for longer to allow it to reach the effective temperature.
A: The length of time to pasturize extends into the boiling time at higher elevations. The length of time at elevation boiling will transfer to the low temp pastuization tactics. To confirm. You MUST boil water and for a longer period over 10,000 feet, I have several friends that got either giardia or crypto over 10,000 feet just bringing water to a boil. You must prolong the boil, hence if you travel over 10,000 feet expect to carry a ton of fuel if you use fuel to purify water. If you aren't using it to purify water and only want to cook with it boil it for at least a few minutes.
A: The boiling point of water is always reduced at altitude. https://www.thespruceeats.com/boiling-points-of-water-1328760
