The Magic of the Primes

In mathematics, we start with the natural numbers (or more simply the 'counting' numbers) and learn how to count, starting with 1 and moving up the positive number line. But something special about counting numbers is usually overlooked- primes. 
Looking at the naturals, we have {1, 2, 3, 4, 5, 6, 7, ... } 
all the way to infinity. 
Now if you pick a number, any number, and analyze it, you can see its basic properties such as its factors or its multiples. Let's take the number 4 for example. 
4 is a multiple of 2, which means it can be divided by a number other than itself and 1. We write "2|4" meaning "two divides four." 
Obviously, most other numbers have these factors and are built on them. But there is a type of number that you may be familiar with but not realize its significance in mathematics. One of the most interesting things in mathematics, though a basic concept, is prime numbers. I'll show you why.
Let's take the number 5. Does it have a divisor? Well, other than 5 and 1, 5 cannot be divided by any other number without it becoming a real number. Why is this significant, you may ask? 
Well, just like the number 2 that divided 4 in the previous example, 5 is a similar divisor to other numbers. For example:
5(2) = 10; 5(4) = 20. 
But we cannot break down 5 as we can its multiples. This goes with any prime number in the set of all integers. Look at 7 or 9:
7(2) = 14; 9(3) = 27 = (3)(3)(3)
7, 2, and 3 are all prime numbers. So that 27 above is broken down into primes. That 14 above is broken down into primes. This is significant, because it means that whatever number you pick from the set of integers it can be broken down into primes numbers. 
200 = 2(100) = 2(25)(4) = (2)(2)(2)(5)(5)
Primes are the building blocks of numbers, which is amazing! 
Now the question might arise: "Are there an infinite amount of primes?"
The answer is YES!!!
Let's say we have the set of all primes P, written
P:= {2, 3, 5, 7, 11, 13, 17, ...}
and this set is finite. Fundamentally, we know that p1=2<p2=3<p3=5<...<pn , were pn is the nth prime. 
Say a number R = p1p2p3 ... pn. Let q = R+1. 
If q is prime, then there is at least one more prime added to the list
If q is not prime, then a prime factor p divides q. 
If p is in the set, then it would divide R, but p divides R+1 which is q,
and if p divides R and q then it divides the difference, i.e. (R+1)-R, or simply 1. 
But p cannot divide 1, since no prime number divides the number 1. 
Thus, this is a contradiction, meaning p cannot be on our list of primes. 
This implies that one more prime number exists beyond the primes listed. 
There it is, in all of its glory. This is Euclid's Theorem, for those of you who are interested. (He also did some awesome work on geometry and number theory- one of my favorite subjects!)
So now we know that there are infinite number of primes, and that prime numbers are the building blocks of the integers. 
ANOTHER COOL FACT ABOUT PRIMES: Cicadas (the big flying bugs) emerge in prime years to have an advantage over predators. One region of cicadas breeds every say 7 years while another every 13. This is advantageous, because it means that while one group of cicadas may be vulnerable to predators, another group of cicadas is breeding for a prime amount of years. This creates a slow but strong population quantity as prime numbers are each unique. 
I hope I explained that clearly. There is more about it on wikipedia or elsewhere! 


Michael H.

Prealgebra, Algebra I and II, Calculus, Geometry, Trig, SAT/ACT Math

20+ hours
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