The largest prime number [1]—discovered by a group of scientists led by Patrick Laroche in 2018—drew considerable interest [2]. To better describe the greatness of this number with 24,862,048 decimal digits, we can give the following example: if we print this number on both sides of A4 papers with standard font sizes, the height of these papers placed on top of each other would be 83 cm. Why did these scientists go to such lengths to make this discovery? Why did such a discovery receive so many awards? These questions will lead us on a voyage into the mysterious world of mathematics.

History of prime numbers

A prime number is a positive number which is evenly divisible only by itself and 1. They start with 2, are followed by 3, 5, and 7, and seem to follow no discernible pattern after that. They have always compelled attention as many scientists have tried to come up with a formula to produce them for centuries. According to the available resources, the oldest study on the matter started with Greek polymath Eratosthenes (BC 276-195) and his famous sieve [3]. The sieve of Eratosthenes systematically eliminates composite numbers—the numbers that are not prime—and defines the remaining numbers as prime, but it is very hard to use for large numbers. In comparison, the Mersenne Theorem [4], developed by French scientist Marin Mersenne (1588-1648) and formulated as  (n is a natural number), helps us to calculate many prime numbers. Although it was later found that 2047 () is composite, all prime numbers calculated by this formula came to be referred to as Mersenne primes. The largest prime number discovered in 2018, too, is a Mersenne prime that satisfies this formula.

Another French mathematician, Pierre de Fermat (1607-1665), adopted a similar approach to Mersenne’s and stated that the formula (m is a natural number) can give us prime numbers—but exceptions to this formula have demonstrated that the formula may not always produce primes [5]. These numbers were called Fermat numbers and worked better than Mersenne numbers [6].

Today, it is generally accepted that there is no formula that can give you all prime numbers. Therefore, mathematicians focus on methods to check whether a number is prime. Researchers have come up with various primality tests including the Fermat, Miller-Rabin, AKS, and Lucas-Lehmer to determine if very large numbers are prime.

In addition to prime numbers, mathematicians also refer to the numbers that are relatively prime, i.e., coprime numbers. Take 21 and 10. Although both of these numbers are not prime, they are relatively prime as the two numbers have only one common divisor, 1. Many composite numbers can be coprime with any other number.

Prime numbers and encryption

We use prime numbers every day without even realizing it. It is possible to give examples of how they are used in many fields including physics, engineering, coding, and diverse technological applications. Cryptography or the practice of cryptography is one of the areas where primes are frequently employed. Deriving from ancient Greek words of “kryptós” meaning “hidden or secret” and “graphein” meaning “to write,” cryptography refers to the study of techniques for secure communication. One of the early applications of cryptography involves the use of the Caesar cipher [7].

Caesar cipher.

The Caesar cipher employs systematic encryption; letters are assigned to numbers:

A=0, B=1, C=2 … Z=29

Then, the message is encoded:

S E L A M

21 5 14 0 15

Using a “key” (for instance, adding 5 to each number), the message is encrypted before it is sent:

26 10 19 5 20

The recipient of the message decrypts it using the rules of modular arithmetic (in this case, by subtracting 5 from the numbers). What is crucial here is to be able to communicate the key to the recipient in a secure manner. During the Second World War, German scientists aware of this problem used the Enigma system, where the key would be changed every day during the war [8]. The British used machines developed by mathematician and logician Alan Turing (1912–1954) to decode the encrypted messages despite the fact that keys were changed on a daily basis, thereby changing the course of the war.

Enigma cipher machine used by Germans.

Prime numbers come to our rescue in solving the security problem in systematic encryption. The communication between the sender and the receiver is established using the method called “asymmetric encryption” without using any key. In 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman developed the RSA encryption method—the name deriving from their surnames—in which communication is secured between the sender and the receiver by using two primes and a third number obtained by multiplying the primes [9]. The first of these primes is called “open key” which is publicly available. The second prime is the “special key” known to each user. The third number, which may not be prime, is a prime obtained by multiplying the first two numbers. Through special operations, the system matches these prime numbers and the key is encrypted. For example, when you insert your bank card to an ATM machine, your card gives the bank’s publicly available key (prime number) to the machine. Then, the machine asks you to enter your PIN, which does not have to be a prime number, for secure communication with your bank. When you enter your PIN correctly, the system has access to the private key through some special operations in the background and allows you to perform transactions related to your account. This process relies on complex algorithms for ensuring security. Whichever algorithm is used, all encryption and decryption operations are performed using prime numbers. The larger the prime number, the harder it will be to break the code.

As is seen, we frequently use prime numbers in our daily life—often without even being aware of them. There may be undiscovered properties and novel areas of the use of prime numbers.

References

1. “Great Internet Mersenne Prime Search”, www.mersenne.org

2. “Largest known prime number”, en.wikipedia.org/wiki/Largest_known_prime_number

3. “Sieve of Eratosthenes (Eratosthenes’in Eleği)”, tolpp.com/sieve-of-eratosthenes-eratosthenesin-elegi/

4. “Marin Mersenne”, www.mathematik.ch/mathematiker/mersenne.html

5. “Pierre de Fermat”, www.spektrum.de/wissen/pierre-fermat-1607-1608/962953

6. →prime
→prime
→prime
→prime
→composite
→prime
The sequence of Mersenne primes in the binary number system.
→ prime
→ prime
→prime
→prime
→prime
→composite
The sequence of Fermat primes in the binary number system.

7. “Caesar cipher”, en.wikipedia.org/wiki/Caesar_cipher; “Caesar Cipher Exploration”, en.khanacademy.org/computing/computer-science/cryptography/crypt/pi/caesar-cipher-exploration

8. “Enigma machine”, en.wikipedia.org/wiki/Enigma_machine

9. “What is RSA encryption and how does it work?”, www.comparitech.com/blog/information-security/rsa-encryption/

My favorite number is 7. It’s a beautiful number and, most importantly, it is a prime number. I emphasize it being prime because I am fascinated by prime numbers. A prime number is a positive number that is only divisible by 1 and itself. Math teachers emphasize teaching prime numbers because prime numbers lie at the very heart of mathematics. But prime numbers are actually not only of purely mathematical interest. When you answer your phone, you are using prime numbers. When you buy something on Amazon, prime numbers are protecting you from thieves. Some animals and fruits love prime numbers also. If you keep reading, you will see that a lot of algorithms develop from number theory, using it for communication security and for the transmission and compression of information.

Prime numbers cannot be evenly divided into smaller integers. Hmmm, so we can take half of all possible numbers off the table right away. Yes, we can! Even numbers, except for 2, cannot be prime. All multiples of three, four, five, six, seven, eight, and so on can also not be prime. At this point you might think, “Hey, there are no prime numbers then.” However, this is not true. There are an infinite number of primes. Euclid proved the theorem many years ago.

Theorem: There are infinitely many primes.

Proof: Suppose that p1=2 < p2 = 3 < … < pr are all of the primes.

Let P = p1 x p2 x … x pr+1 and let p be a prime dividing P; then p cannot be any of p1, p2, …, pr, otherwise p would divide the difference P – p1 x p2 …pr=1, which is impossible. So, this prime p is still another prime, and p1, p2, …, pr would not be all of the primes.

I know the theorem seems complicated. But the basic idea of the proof is that, if there were only finitely many primes, we could have a list of all of those prime numbers. For example, let’s say we only know 2, 3, 5, and 7 as prime. Now if you multiply them all together and add 1, you will get a new number, 211, that isn’t divisible by any of the numbers except for 1 and itself. So, we get a new prime number for our list. You can do this operation infinitely and every time you will get a new prime. So, if somebody asks you what is the largest prime, you know the answer. “I don’t know! They are infinite.”

Assumption: There is no real formula which can be used to find all the prime numbers.

When you start writing the prime numbers in order, at the beginning you will see that there are some pairs of prime numbers that have just one number between them. We call them “Twin Primes.” For example: 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31.

But you will realize that they are less frequent as you go on. There is a conjecture about this.

The twin primes conjecture: There are infinitely many pairs of twin primes among the infinitely many prime numbers.

Furthermore, having no divisor is not the only thing that makes prime numbers so unique and cool. For instance, a whole number line can be produced using nothing but primes. In other words, every other number can be found by multiplying primes together. Thus, we can easily say that primes are the atoms of the number sequence and all other numbers are built from primes. And then from numbers you get mathematics and from mathematics you get the whole of science. Great mathematicians have shown that:

absolutely any whole number can be expressed as a product of primes, only primes, and nothing else.

Mathematicians call this the unique prime factorization theorem. This theory is coming from number theory, the Fundamental Theorem of Arithmetic. And understanding the fundamentals of arithmetic is the first key to unraveling the mysteries of the primes.

Fundamental Theorem of Arithmetic: Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

For example:

To get 666, try this one: 2 x 3 x 3 x 37

What about 1,234,567,890? It is not that hard.

2 x 3 x 3 x 5 x 3607 x 3803

Interesting fact: There are people who are really good at calculating in their heads. Don’t worry if you cannot, for a lot of mathematicians cannot calculate at all. They just use their fingers or do things on paper.

Basically, we are pulling apart any number into two numbers, then pulling those apart into two numbers if possible, and so on… When no further factoring can be done, all numbers left over are primes. We will eventually be left only with primes. The terms in the product are called prime factors. And do not forget that the same prime factor may occur more than once; the example above and below has two copies of the prime factor of 2:

1,234,567,890= 2 x 3 x 5 x 3607 x 3803

Note: When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number.

A particularly beautiful prime number is Belphegor’s Prime, 1000000000000066600000000000001.

That’s a 1, followed by 13 zeros, followed by a 666, followed by 13 more zeros, followed by a closing 1. Can such beauty really be a coincidence?

What about the largest prime number? I found that as of January 2018, the largest known prime number is 2^77,232,917 − 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search.

When I first checked the biggest prime number, I noticed that finding them is unpredictable. And being unpredictable makes prime numbers mysterious. Normally, special numbers should not occur randomly. All the mathematicians that I know have always believed that. And luckily, researchers from Stanford checked the first 100 million primes, searching for a pattern. And they found one.

Here is a great piece of information: Prime numbers have a peculiar dislike for other would-be primes that end in the same digit.

I wondered: Why do some smart people spend their lives trying to find numbers which are not going to be divisible by any other number? Why?

The reason was simple: To save our lives.

It is all about “encryption.” Prime numbers mean encryption. This makes primes vitally important to communications. Most modern computer cryptography works by using the prime factors of large numbers. Prime numbers also played an important part in the secret spy codes during World War II.

Encryption means that communication is scrambled in some way. It is based on the science of cryptography, which has been used as long as humans have wanted to keep information secret. If you watch the movie, “The Imitation Game,” you know that, during World War II the biggest users of cryptography were governments, particularly for military purposes. And during the Cold War, the U.S. and Russia both tried to keep the other from learning defense secrets about missile and rocket developments and other military secrets.

Now we live in the digital age, and we put more information online, like keeping our photos or messages on iCloud or putting credit card information on Amazon; like logging into our bank account or sending a very important email to our professor or having a private conversation with our best friends on the phone. All these financial transactions and other sensitive details are not just important to us, but also to thieves. We have to stop people stealing our info. And that means we have to respect prime numbers and constantly try to find new prime numbers.

How has the mathematical logic behind them resulted in vital applications in the modern world? The reason is simple. You could do a prime factoring for a 4-digit number during a test, but what if somebody gives you a 600-digit number or 1000-digit number? Your algorithm definitely will not work. Because even the most powerful and advanced ultra-super computers cannot do prime factorization for a long time. Not seeing your family for a week could be a long time, but here I am trying to say that “long time” is equal to the age of the universe. It may sound absurd, but it is reality.

For example, no one has been able to find the two primes which divide the following 617-digit number:

25,195,908,475,657,893,494,027,183,240,048,398,571,429,282,126,204,032,027,777,137,836,043,662,020,707,595,556,264,018,525,880,784,406,918,290,641,249,515,082,189,298,559,149,176,184,502,808,489,120,072,844,992,687,392,807,287,776,735,971,418,347,270,261,896,375,014,971,824,691,165,077,613,379,859,095,700,097,330,459,748,808,428,401,797,429,100,642,458,691,817,195,118,746,121,515,172,654,632,282,216,869,987,549,182,422,433,637,259,085,141,865,462,043,576,798,423,387,184,774,447,920,739,934,236,584,823,824,281,198,163,815,010,674,810,451,660,377,306,056,201,619,676,256,133,844,143,603,833,904,414,952,634,432,190,114,657,544,454,178,424,020,924,616,515,723,350,778,707,749,817,125,772,467,962,926,386,356,373,289,912,154,831,438,167,899,885,040,445,364,023,527,381,951,378,636,564,391,212,010,397,122,822,120,720,357

Let me give you another example. Every time you send your credit card number to Amazon, you are depending on primes to keep your details secret. To encode your credit card number, your computer receives a public number like X from the website, which it uses to perform a calculation with your credit card number. This scrambles your details so that the encoded message can be sent across the internet. But to decode the message the website uses the primes which divide X to undo the calculation. Although X is public, the primes which divide X are the secret keys which unlock the secret.

To put it briefly, prime numbers protect your money and privacy from being stolen. A prime number is absolutely essential to modern computer security.

One of the amazing things about prime numbers is how their presence can be felt in nature.

I am sure that you like the sound of cicadas during the summer. However, people in Nashville do not have a chance to listen to the harmony of the cicadas every year. The last time they had a chance to listen to the cicadas was 2011. Nashville’s forests will be quiet for 13 or 17 years. 13 or 17? But they are not divisible by any number except for 1 and itself. Yes, they are prime numbers.

According to scientific research, the cicadas are mathematical bugs. They appear periodically but only emerge after a prime number of years. They use prime numbers to come out of their burrows and lay eggs. Cicadas only leave their burrows in intervals of 7, 13, or 17 years. You may think that the cicadas choose these numbers randomly. You would be wrong. There are no cicadas with 8, 10, 12, 15, or 20-year life cycles. If you start looking at these cicadas through the mathematician’s eyes, you will see that the pattern begins to emerge. It is also a beautiful chance to understand how perfectly created is nature.

13 and 17 are both indivisible, and this gives the cicadas an advantage, as primes are helpful in avoiding other animals with periodic behavior. But, how? Suppose that a predator appears every four years in the forest. Then a cicada with a six-year life cycle will coincide with the predator every 12 years. Yes, because the least common multiple of 4 and 6 is 12. But if a cicada emerges every thirteen years, it will only face a peak predator population once every fifty-two years. (4 x 13). And that gives it an enormous advantage.

In other words, these insects use prime numbers to ensure their survival. The cryptography that keeps our lives secure when we are online uses the same numbers that protect the cicadas.

Mathematics is a world of mystery. And at the core of it lie the prime numbers.