A very very light introduction to Number Theory

This little article is a very very light introduction to Number Theory for those motivated to attempt RMO/INMO. On the other hand, it is a light introduction for anyone interested in beginning Number Theory. 

Number Theory is the study of the set of positive whole numbers 1,2,3,4,5,6,7…

which are often called the set of natural numbers. We will especially want to study the relationships between the different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here are some familiar and not so familiar examples:

odd numbers {1,3,5,7,9,11…}

even numbers {2,4,6,8,10,…}

square numbers {1,4,9,16,25,36,…}

cube numbers {1,8,27,64,125,…}

prime numbers {2,3,5,7,11,13,17,19,23,29,31…}

composite numbers {4,6,8,9,10,12,14,15,16…}

1 (modulo 4) {1,5,9,13,17,21,25…}

3 (modulo 4) {3,7,11,15,19,23,27…}

triangular numbers {1,3,6,10,15,21…}

perfect numbers {6,28,496,,,,}

Fibonacci numbers {1,1,2,3,5,8,13,21…}

Many of these types of numbers are undoubtedly already known to you. Others, such as the “modulo 4” numbers, may not be familiar. A number is said to congruent to 1(modulo 4) if it leaves a remainder of 1 when divided by 4, and similar for the 3(modulo 4) numbers. A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the original number. Thus, the numbers dividing 6 are 1,2 and 3, and 1+2+3=6. Similarly, the divisors of 28 are 1,2,4,7 and 14, and

1+2+4+7+14=28.

We will encounter all these types of numbers, and many others, in our excursion through the Theory of Numbers.

Some Typical Number Theoretic Questions.

The main goal of number theory is to discover interesting and unexpected relationships  between different sorts of numbers and to prove that these relationships are true. In this section, we will describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day.

Sum of SquaresI. Can the sum of two squares be a square? The answer is clearly “YES”; for example, 3^{2}+4^{2}=5^{2} and

5^{2}+12^{2}=13^{2} These are examples of Pythagorean triples. We will describe all Pythagorean triples in a later blog.

Sums of Higher Powers. Can the sum of two cubes be a cube? Can the sum of two fourth powers be a fourth power? In general, can the sum of two nth powers be an nth power? The answer is NO. This famous problem, called Fermat’s Last Theorem, was first posed by Pierre de Fermat in the seventeenth century, but was not completely solved until 1994 by Andrew Wiles. Wiles’s proof uses sophisticated mathematical techniques that we will not be able to describe in detail. But, we can show (in some later blog) that no fourth power is a sum of two fourth powers.

Infinitude of Primes. A prime number is a number p whose only factors are 1 and p.

  • Are there infinitely many prime numbers?
  • Are there infinitely many primes that are 1 modulo 4 numbers?
  • Are there infinitely many primes that are 3 modulo 4 numbers?

The answer to all these questions is “YES”. We will prove these facts later in some blog.

Sum of Squares II. Which numbers are sums of two squares? It often turns out that questions of this sort are easier to answer first for primes, so we ask which (odd) prime numbers are a sum of two squares. For example,

3=NO;

5=1^{2}+2^{2}

7=NO

11=NO

13=2^{2}+3^{2}

17=1^{2}+4^{2}

19=NO

23=NO

29=2^{2}+5^{2}

31=NO

37=1^{2}+6^{2}

Do you see a pattern? Possibly not, since this is only a short list, but a longer list leads to the conjecture that p is a sum of two squares if it is congruent to 1(modulo 4). In other words, p is a sum of two squares, if it leaves a remainder of 1 when divided by 4, and it is not a sum of two squares if it leaves a remainder of 3.

Number Shapes. The square numbers are the numbers 1,4,9,16, …that can be arranged in the shape of a square. The triangular numbers are the numbers 1,3,6,10,… that can be arranged in the shape of a triangle.

A natural question to ask is whether there are any triangular numbers that are also square numbers (other than 1). The answer is YES; the smallest example being

36=6^{2}=1+2+3+4+5+6+7+8

So, we might ask whether there are more examples and, if som are there infinitely many ? To search for examples, the following formula is helpful:

1+2+3+4+...+(n-1)+n=n(n+1)/2

Twin Primes. In the list of primes, it is sometimes true that consecutive odd numbers are both prime. We have written the twin prime pairs less than 100:

(3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61), (71,73)

Are there infinitely many twin primes? That is, are there infinitely many prime numbers p such that p+2 and  is also a prime? At present, no one knows the answer to this question.

Primes of the Form N^{2}+1

If we list the numbers of the form N^{2}+1 taking N=1,2,3,…we find that some of them are prime. Of course, if N is odd, then N^{2}+1 even, so it won’t be prime unless N=1. So, it’s really only interesting to take even values of N. We have highlighted the primes in the following list:

2^{2}+1=5

4^{2}+1=17

6^{2}+1=37

8^{2}+1=65=5.13

10^{2}+1=101

12^{2}+1=145=5.29

14^{2}+1=197

16^{2}+1=257

18^{2}+1=325=5^{2}.13

20^{2}+1=401

It looks like there are quite a few prime values, but if you take larger values of N you will find that they  become much rarer. So, we ask whether there are infinitely many primes of the form N^{2}+1. Again, no one presently knows the answer to this question.

We have now seem some of the types of questions that are studied in the Theory of Numbers. How does one attempt to answer these questions? The answer is that Number Theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part, one tries to devise an argument that gives a conclusive answer to the questions. In summary, here are the steps to follow:

1) Accumulate data, usually numerical, but sometimes more abstract in nature.

2) Examine the data and try to find patterns and relationships.

3) Formulate conjectures (that is, guesses) that explain the patterns and relationships. These are frequently given by formulae.

4) Test your conjectures by collecting additional data and checking whether the new information fits your conjectures.

5) Devise an argument (that is, a proof) that your conjectures are correct.

All five steps are important in number theory and in mathematics. More generally, the scientific method always involves at least the first four steps. Be wary of any purported “scientist” who claims to have “proved” something using only the first three. Given any collection of data, it’s generally not too difficult to devise numerous explanations. The true test of a scientific theory is its ability to predict the outcome of experiments that have not yet taken place. In other words, a scientific theory only becomes plausible when it has been tested against new data. This is true of all real science. In mathematics, one requires the further step of a proof, that is, a logical sequence of assertions, starting from known facts and ending at the desired statement.

More later…

Nalin Pithwa

 

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