Wilson Primes

By Anupum Pant

Thanks to the guys at Numberphile for introducing me to Wilson primes. Although the piece of information that describes Wilson primes itself has more or less no practical use, I still think it’s a good thing to know.

The first thing you need to know is that all prime numbers follow this rule – If you take a prime number P and put it in the following equation you get a number that is perfectly divisible by the prime number P.

The equation: (P − 1)! + 1 = Q

Note: ! is a sign used for factorial. That means P! is equal to the product of all natural numbers smaller or equal to P. So, for example, 3! = 3 X 2 X 1

This rule is valid for all prime numbers and no composite numbers follow it. So, for instance, if you take a composite number for P, the number you get after you put it in the above equation is never divisible by the number itself. This is called the Wilson’s theorem.

Wilson primes (P) are a few special numbers which can divide Q in the equation above two times. So, for example, since 5 is a Wilson prime, you get 25 if you put it in the equation above. And 25 can be divided perfectly by 5 once, and the result (quotient 5) can be divided again by 5 to get a whole number.

Now, for Wilson primes here’s the deal – 5, 13 and 563 are Wilson Primes. And a very interesting thing to note here is that, in spite of all the computing technology we have in the world, these are the only three Wilson primes we know yet.

Mathematicians are pretty certain that there are several other Wilson primes waiting to get discovered, probably infinitely many. But one thing is for sure, below the number 20,000,000,000,000 5. 13 and 563 are the only three which exist.

The Tallest Mountain in Our Solar System

By Anupum Pant

Right here on earth there are really tall mountains. Mount Everest is the highest peak and then there’s Mauna Kea in Hawaii which is supposed to be the tallest. Yes, even taller than the Mt. Everest. To add to it, there’s one highest unclimbed mountain – Gangkhar Puensum – in Bhutan.

If we zoom out a little and put the whole solar system in our radar, things change. Mt Everest or even Mauna Kea are no where near the tallest mountains we have in our solar system. For instance, Olympus Mons, a shield volcano has, for a long time, been considered the highest peak in our solar system.

This is how it compares with mount Everest, for example. The peak of  Mount Everest measures 8,848 meters. It’s absolutely huge. And yet, Olympus Mons on Mars is about 2.5 times higher! It measures about 22 kilometres in height. This image clearly shows how it compares with our tallest and highest mountains…

Olympus_Mons_Side_View.svg

And yet again, even Olympus Mons, which has had the title of the tallest mountain in our solar system for several years, is believed to be no longer the tallest one.

A recently discovered peak in a proto-planet called Vesta is probably now the tallest mountain in our solar system. However, since this one – Mount Rheasilvia – is estimated to be only a few 100 meters taller than Olympus Mons, it has not very clearly dethroned Olympus Mons. Still, the data is pretty solid and can be trusted.

Rheasilvia was a peak known to researchers since 1997. But it was in 2011, when the Dawn spacecraft passed it, the data became really clear.

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The Highest Unclimbed Mountain

By Anupum Pant

Gangkhar Puensum, meaning three mountain siblings, is the tallest mountain in Bhutan with an elevation of 7,570 meters. Since the 80s several attempts have been made to climb this mountain – a part of which lies in Bhutan and the other part in Tibet. None of the attempts have ever been successful.

However in the year 1999, a team of climbers from Japan, after a protracted attempt to get a permit, were able to reach the top of one of the three peaks – Liankang Kangri – from the Chinese side of the mountain. Later, protests from local people in Bhutan made them stop.

So technically, the highest peak has never been climbed by anyone till date. Gangkhar Puensum remains the highest unclimbed mountain. The reason mostly is because obtaining a permit to climb it is almost impossible. It is prohibited by the government of Bhutan.

The prohibition by the government has mostly to do with the lack of rescue services at that place, and due to the local belief which considers the peak sacred – a home to holy spirits.

Gangkhar Puensum is certainly one of the uncharted mysterious places in the world where no one has gone and probably never will.