# What are Prime Numbers?

**Meet the Numeric Celebrity – Prime Numbers**

That’s right, primes are quite the celebrity and not just in Hollywood movies. 😉 Here’s an excerpt of primes being featured in the movie, “Contact” where scientists discover an alien signal composed of… yes you’ve guessed it, prime numbers!

As seen in the movie, the aliens chose to send a long string of prime numbers to prove that their message was intelligent and not of natural origin. So why use prime numbers? What so special about primes?

**What are prime numbers?**

Primes are the building blocks of all numbers. Think of prime numbers as atoms, just like in chemistry where we say that a water molecule is formed from two hydrogen atoms and one oxygen atom (notated as H_{2}O). Likewise, the number 12 is the product of the prime factors 2×2×3 (notated as 2^{2} 3). So just like water can be decomposed into hydrogen and oxygen, all numbers can be decomposed into primes. Here are a few more examples:

- 8 = 2×2×2 = 2
^{3} - 20 = 2×2×5 =2
^{2}5 - 180 = 2×2×3×3×5 = 2
^{2}3^{2}5

This process of decomposing a number into its prime factors is called *prime factorisation* (a topic to be left for another time). Like atoms, prime numbers can’t be decomposed further or rather can’t be divided further, like 2, 3, 5, 7, etc. In other words, **prime numbers are only divisible by 1 and itself**, and a number that has more than 2 factors is known as a composite number. For example:

**2**= 1×2 (2 factors only) →**Prime****3**= 1×3 (2 factors only) →**Prime****4**= 1×4 or 2×2 (3 factors) →**Composite****5**= 1×5 (2 factors only) →**Prime****6**= 1×6 or 2×3 (4 factors) →**Composite**

At this point, you may be wondering – what about the number 1? Is it prime? Well, 1 is somewhat of a special case. If you think about it, 1 = 1×1×1×1×1… and this is where things get a little crazy. If you were to just consider the number of factors 1 has, it’s 1, which is also itself! So… is it prime? There is certainly a little more than meets the eye. 1 used to be prime, but it’s no longer prime. Haha… and the story of primes continue to unravel. To find out more, watch the short video below, where James Grime the Numberphile, concisely explains the Fundamental Theorem of Arithmetic (don’t worry, it’s just a fancy name – watch the video and all will become clear) and how it applies to 1 and prime numbers.

Cool right? Okay, so we now know that 1 is neither prime nor composite. It’s just the lonely one. Awww… poor 1. 😢

Well, is that all there is to prime numbers? Far from it! Here are a few more observations and interesting facts about prime numbers:

- I’m sure you’ve noticed this. 2 is the only even number that is prime. The rest of the prime numbers are odd.
- As numbers get larger, primes become less frequent and twin primes (see below) get even more rare.
- In any case, we’ll never run out of prime numbers, as they are infinite. Any idea what’s the largest prime number ever discovered to date? Watch the final video below to find out.
- Twin primes are pairs of primes that differ by two. The first twin primes are {3,5}, followed by {5,7}, {11, 13} and so on. It has been conjectured (meaning it’s never been proven) that there are infinitely many twin primes. This is known as the twin prime conjecture, a.k.a. Euclid’s twin prime conjecture.
- Prime factorisation is hard work and when numbers get extremely large, you can imagine how tedious and slow it’ll be. ?
- On top of this, primes do not have a pattern we can easily decipher, meaning there is no easy way to tell when the next prime number will appear. But that’s actually a blessing in disguise. Why? Read on to find out more.

**Why are prime numbers so important?**

Did you know that prime numbers are worth billions of dollars? 😲 Why are prime numbers so valuable to organisations, government agencies and companies like Apple, Google, eBay or Visa? Wondering how numbers can be worth even a single cent? Well, though prime numbers have little value in themselves, they are used in every credit/debit card transaction, including ATMs, online payments and even trading (e.g. stocks and shares) transactions totalling billions of dollars every day. In fact, prime numbers power the mathematics behind the cryptography (used for cyber security) of your WIFI connections, email accounts, blogs, Facebook, Twitter, etc.

To find out how primes combined with the difficulty of factoring large numbers are used to protect and secure our emails and payment transactions, please watch the short video below.

Aren’t prime numbers just fascinating? As Carl Sagan, author of the science fiction novel, “Contact” so eloquently pointed out – there is a certain importance to the status of prime numbers as the most fundamental building block of all numbers, which are in turn themselves the building blocks that help us understand our universe. 🤔 Regardless of how an advance alien life form may think or look like, one thing is for certain, if it understands the world around it, it most certainly understands the concept of primes.

Hope you found this article insightful and educational. Happy maths! 😁

Sponge ME, Maths Tuition (Singapore)

Oh… and if you are interested to find out what is the largest prime number ever discovered to date (Jan 2016), here is Matt Parker on the latest Mersenne Prime that holds the envious world record. Who knows? Maybe you might be the next record breaker for finding the “world’s largest prime”. Find out how it’s done and more in this video. Enjoy! 😉