By Lou Feng
On Mar. 21, Pittsburgh had the honor of hosting Fields Medalist James A. Maynard at the Mathematical Horizons Lecture Series. The inaugural event took place at 3:30 p.m. in the Frick Fine Arts Building, with a reception following Professor Maynard’s lecture until 5:30 p.m. Swarms of college students, eager math-minded youngins, and olds donning glasses and carrying textbooks flooded through the doors of the auditorium to hear the tales of primes and their twins.
The Twin Prime Conjecture states that “there are infinitely many primes p such that p+2 is also prime.” Some twin primes are familiar to us: five and seven, 11 and 13, etc. The question of whether this separation of the two continues on forever remains unanswered, hence the name “conjecture.” We are, however, able to place bounds on the distances between primes. Professor Yitang Zhang in 2013 presented a proof which demonstrated that there exists some number N less than 70 million for which there are infinitely many primes whose difference is N. Further efforts on Zhang’s proof have lowered this N down to 246, but, since 2014, the bound remains unaltered.
In a similar vein, the Fundamental Theorem of Arithmetic states that “every integer greater than one can be represented uniquely as a product of prime numbers, up to the order of the factors.” This is the basis of prime factorization.
For the less number-theoretic trained, these facts about primes may seem distant from your daily lives. Professor Maynard argues that, more than just the favorite pastime of mathematicians, primes play a fundamental role in computer science and technological advancement.
Suppose you are surfing the web when you suddenly stumble upon an ad for a painting of a toad wearing a fedora and eating a croissant. Clearly, this is a necessary purchase. You go to enter in your credit card details on the site, then hesitate: how do I know my information is protected? In comes the power of primes.
Computing power has allowed us to easily multiply numbers with hundreds and thousands of digits together in mere minutes, even seconds. Factoring, on the other hand, proves far more difficult. Given a number like 15, it’s easy to know its prime factors, three and five. But if I asked you to give me the prime factors of a 174 digit number, it may take you a little while. In fact, correctly giving the two prime factors of this monster of a number in 2003 would have earned you $10,000 as a part of the RSA (Rivest–Shamir–Adleman) factoring challenge. RSA is an example of a public-key encryption algorithm that uses the fact that multiplying is easy, but factoring is hard, to securely store information on the internet. Even if a hacker were to get their hands on the result of two primes multiplied together, they would struggle to determine its factors. In some way, primes are protecting you.
Professor Maynard encourages old and young, nerd and jock, to all look to maths “in their prime.”
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