What is number theory?

Anyone who has ever fallen in love can tell you it is the little things about the other person that make them attractive: the way they do their hair, the peculiarities of how they write, the way they sing a certain part of a song out of key every time... Such details come to define us. They trace the outlines of our personalities, and, to the observant eye, they illuminate true beauty. In the eyes of some, there is no finer beauty than that found in mathematics. Some look at numbers and, just as you’d never define your beloved human solely based on their eye color, the math lover sees beyond the mere function of numbers. The likes of 1, 2, and 3, turn into something more sublime than simple carriers of information. The math lover looks for questions, patterns, and proofs as to why simple equations like 1+1 equals 2. This is essentially what number theory is: the study of subtle and radical relationships between and among numbers.

Euclid of Alexandria was a key contributor to the foundation of number theory. He developed and proved the concept of infinite primes, which eventually became a key notion in the field of number theory. As most of us know, prime numbers are integers that are greater than one and have only two factors - one and themselves. Euclid hypothesized that there was an infinite number of primes, which he then sought to prove. He did this through a method known as proof by contradiction. Essentially, Euclid assumed that there was a finite number of primes, and disproved that concept, which leads to the conclusion that an infinite number of primes exist. This concept of infinite primes would become one of the most foundational ideas of number theory.

While number theory may seem like an abstract concept that is only used by those who work in the math field, it has many practical applications. For instance, many companies use encryption to protect their data, which is a concept that stems from the number theory. Oftentimes, encryption is done using basic primes of a very large number. The basic primes of a number are obtained when the integer is broken down into its prime factors, which you may know by its other name: prime factorization. Each number has a unique set of prime factors, and by using the basic primes of large numbers, companies can ensure that their data is safely protected.

Now that you know the basics of number theory, try it out yourself. I will give you a sequence of numbers and you have to try and find a pattern, formula, or some sort of relationship between the numbers in order to figure out what the first term of the sequence is. Remember: think creatively, not everything is so black and white. Example: what is the first term in the sequence? ….11, 24, 75, 304

1. 5

2. 8

3. 9

4. 10

Explanation:

Alright, so you decided to take on the example problem and have come searching for the answer. The answer is *drum roll* 10! I hope you got it right. If not, it is okay. Let me explain to you how I went about it. I noticed that the second term, 11, and the third term, 24 could be formed by taking the second term and multiplying it by two and adding two. However, this equation of (n*2+2) does not work for the third number, 24, into the fourth sequence, 75. The equation that does work for these two transitions is, however, (n*3 +3). Taking 24 times 3 and adding 33 to that gets you 75! Therefore, it is visible that there may be a pattern forming in the equations from each preceding number. I finally checked to see if 75 times 4 plus 4 (n*4 +4) gets you to 304, and it did! There you go! You figured out the pattern. Using this pattern, you worked backward to find out that the first term is 10 since 10 times 1 plus 1 (n*1 +1) is 11! Good job, VSN!