A prime number is a natural number greater than 1 that cannot be written as the product of two natural numbers except 1 and itself. In our daily lives, the applications of mathematics are usually the calculation of elementary arithmetic. There is, therefore, no apparent connection between prime numbers and our daily lives. In fact, there is discernible linkage.

Classic Mathematical Challenge
Prime numbers have been a source of mathematical intrigues since time immemorial. Euclid's theorem first proved that "there are infinitely many prime numbers" before the Common Era. Our 2015 Shaw Prize in Mathematical Sciences, Prof Henryk Iwaniec had successfully delved into the characteristics of prime numbers. Iwaniec’s foundational work and breakthroughs in sieve theory and its applications form a large part of this area of mathematics. In 1997, Iwaniec and Canadian Mathematician John Friedlander proved that there are infinitely many prime numbers of the form X2 + Y4. For example, 17 is a prime number and it can be calculated by substituting 1 and 2 into X and Y respectively, that is 12+24=17. Results of this strength had previously been considered out of reach. The research of the two brings a breakthrough in the application of the sieve theory and lays the foundation for further prime numbers investigation such as the research in prime gap by the Chinese mathematician Yitang Zhang.

Prime Numbers in Daily Life
Here comes a question: how do prime numbers relate to our daily lives? In fact, it is hidden stilly in people’s lives. Prime numbers are, actually, the building block of secured encryption. According to 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 (factors), we call it prime factorisation. For example, 60 can be prime factorized and represented as 2 x 2 x 3 x 5. This representation is unique, except for the order of the factors. Prime numbers have some special properties for factorisation, one of them is that it is very hard to prime factorize a large number. For example, if we are given a 7-digit integer 3,526,842, it is very difficult to find out that it can be represented by 2 x 3 x 11 x 53437 and they are all prime numbers. In 1977, using the unique properties of prime numbers, Ron Rivest, Adi Shamir, and Leonard Adleman, three scientists from MIT, derived Rivest-Shamir-Adleman (RSA) cryptosystem. RSA is widely used for secure data transmission and digital business such as internet banking, which many of us rely on daily.

RSA is a kind of asymmetric encryption method. The user needs two keys to decipher encrypted messages. These two keys are "public key" and "private key". "Public key" is shared by the sender and the receiver. "Private key" is held only by the receiver. The principle of the RSA encryption method is to encrypt the original message with a public key when sending a message. After receiving the encrypted message, the receiver needs to decrypt it with the private key to see the original message. Asymmetric encryption ensures that only the sender and receiver can read the message. Even if the message is intercepted, the interceptor cannot decipher and peep into the encrypted message. Besides, it ensures that the message will not be forged by others since the only people who know the two keys are the sender and the recipient.

As at 2018, quantum computers can only successfully hack the 768-bit RSA cypher, but the current commonly used cypher on the Internet is 1024 bits. In some important occasions, RSA cypher can be as long as 2048 bits. Based on security needs, the system provider will also change the encryption method every few months. Therefore, it is believed that RSA will still be the most secure information encryption method in the foreseeable future based on the characteristics of prime numbers.

Prime Numbers in Nature
Besides the application on data security, we may also find prime numbers in other interesting areas. Periodical cicadas, also known as prime cicadas, is a type of cicadas that spend almost the full length of their life span underground. They only emerge on the ground towards the end of their lifecycle to breed. Their lifecycle can be 7, 13 and 17 years which interestingly are all prime numbers. Indeed, in the summer of 2020 a group of cicadas climbed out of the ground in parts of the US after 17 years to complete their mating cycle. Although scientists do not have a firm conclusion on the prime number lifecycles of periodical cicadas, some believe that it is related to the avoidance of their predators. Periodic cicadas are less likely to encounter their natural enemies as they spend years as nymph underground. Moreover, with the synchronised emergence in their millions, the sheer number of periodic cicadas makes good defence for the brood’s overall survival.

Prime number remains an interesting topic in mathematics. More discoveries on properties of prime numbers may derive more theories and applications in the future.

November 2020