Hi Steemicians! Today we will be covering a question from @isa93 on this post over here. Her question concerns prime numbers. Prime numbers are numbers greater than one whose only divisors are 1 or itself (a divisor of a number n is a number k such that n/k is a whole number). So 3,5,7 are examples of primes. @isa93 her (rephrased) question is

Why is 1 not a prime number?

This question is connected to a fundamental result of number theory which gives the building blocks of the whole numbers. We will start with this fundamental result and then work back to the question.

The Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic is a result that answers the question how can you construct whole numbers? This brings us to the question what does construct mean in a mathematical sense? Construction of a number is always in relation to a mathematical operation. Mathematical operations are in this case operations on numbers, so for example addition, multiplication. It is easy to see that if you use the number 1 and addition then you can construct any whole number. For example we have that 5=1+1+1+1+1. So constructing any number with addition is easy. How about if I can only multiply? Now it is time to introduce the fundamental theorem of arithmetic:

Fundamental Theorem of Arithmetic. Every whole number is either a prime or can be written uniquely (up to re- ordering) as primes multiplied by each other

Let us explain this with an example: 24=2 x 2 x 2 x 2 x 3. Observe that 2,3 are prime numbers. The written uniquely (up to re-ordering)-part in the theorem means that we consider 2 x 2 x 2 x 2 x 3 and 3 x 2 x 2 x 2 x 2 the same. So for this case the theorem says that I always need to multiply the same prime numbers the same amount of times to get 24. Here is another example: 72=3 x 3 x 2 x 2 x 2

Informally the theorem says that the prime numbers form the unique building blocks of any whole number under multiplication.

Fundamental Theorem of Arithmetic fails when 1 is prime

So now we get back to the question of @isa93. Suppose for a moment that 1 would be considered a prime number. Let us check whether the fundamental theorem of arithmetic still can be true. Again we look at 24. Then 24= 2 x 2 x 2 x 3=1 x 2 x 2 x 2 x 2 x 3 =1 x 1 x 2 x 2 x 2 x 2 x 3, because we considered 1 a prime the fundamental theorem of arithmetic cannot be true anymore! So because we want the fundamental theorem of arithmetic to be true we define primes such that 1 is not a prime. This seems a bit like cheating....

The relation between definition and theorem

In mathematics you define stuff so that that you can prove awesome things for the stuff that you defined. Sometimes the stuff that you define comes very natural but other times you need to adapt the stuff you define so that you can prove something for it. Of course you can also relate this to real-life things. I can only enjoy drinking a beer if I remove the beercap ;). So that is pretty much the relation between prime numbers and the fundamental theorem of arithmetic. You can only enjoy using the fundamental theorem of arithmetic if you remove 1 from the prime numbers.

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Sources & further reading

Today everyhing is completely original. But if you are interested in a proof of the fundamental theorem of arithmetic you can take a look over here

All figures made using Inkscape it is free to use!

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Thank you!

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below :o)

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