Number Theory is the study of the integers.
Primes
One of the most basic properties of the integers (and one of the most studied) is which ones are prime. A prime number is one which is only divisible by itself and by 1. The first few prime numbers are
2,3,5,7,11,13,17,19,23,29,31,37…
Diophantine Equations
Diophantine equations are one of the most studied parts of number theory. Most of the subjects below are Diophantine equations underneath.
Taxicab Numbers
G. H. Hardy went to visit his friend Sinrivasa Ramanujan the famous Indian mathematician in the hospital. While trying to start a conversation he mentioned that he had come in taxi number 1729 and remarked what a boring number it was. Ramanujan replied that it was not boring at all, that it was the smallest number that could be written as the sum of two cubes in two different ways.
123+13=1729; 93+103=1729 This was the beginning of taxicab numbers. The general form for a taxicab number
(x,y,z) means the smallest number which can be written as the sum of x, yth powers in z different ways.
Fermat Primes
Fermat Primes are named after Pierre de Fermat. If a number
22n+1 is prime it is a Fermat Prime. The first few Fermat Primes are:
3, 5, 17, 257…
Diophantine Equations
Diophantine Equations are named after the ancient mathematician Diophantus who first introduced and defined the form.
A Diophantine Equation consists of a polynomial which is composed of integers and has integer solutions. Any other solutions are considered invalid.
One of the most famous Diophantine equations is known as Fermat’s Last Theorem.
az+bz=cz has no integer solutions for z>2. This means although a
2+b
2=c
2 has many solutions, (3
2+4
2=5
2 for example) a
3+b
3=c
3 has no integer solutions, a
4+b
4=c
4 has no integer solutions, a
5+b
5=c
5 has no integer solutions, ... etc.
Interesting Numbers
Many numbers are interesting for one reason or another. Eric Friedman has compiled a list of 9999 interesting numbers.
Aliquot Sequence
The Aliquot Sequence for a number is generated by finding the sum of its factors for example
12:16, 15, 9, 4, 3, 1, 0
Ways for an aliquot sequence to end:
- It reaches 0
- It reaches a perfect number which maps to itself.
- It goes into a loop
- It grows indefinitely
Loops of Aliquot sequences are called amicable pairs (Period 2) or Sociable numbers (longer Periods).
Perfect Numbers
Another important type of number is Perfect Numbers
6, 28… These numbers are determined by the sum of their factors (excluding themselves). For
6:1, 2, 3 these numbers add up to six. If the factors of a number add up to less than the number it is called Deficient.
8:1, 2, 4 and 1+2+4=7<8 If the factors add up to more than the number it is called Abundant.
12:1, 2, 3, 4, 6 and 1+2+3+4+6=16>12
Here is a table for the first 20 numbers
| X |
Factors of X |
Sum |
Type of x |
| 1 |
No other factors |
0 |
Deficient |
| 2 |
1 |
1 |
Deficient |
| 3 |
1 |
1 |
Deficient |
| 4 |
1,2 |
3 |
Deficient |
| 5 |
1 |
1 |
Deficient |
| 6 |
1,2,3 |
6 |
Perfect |
| 7 |
1 |
1 |
Deficient |
| 8 |
1,2,4 |
7 |
Deficient |
| 9 |
1,3 |
4 |
Deficient |
| 10 |
1,2,5 |
8 |
Deficient |
| 11 |
1 |
1 |
Deficient |
| 12 |
1,2,3,4,6 |
16 |
Abundant |
| 13 |
1 |
1 |
Deficient |
| 14 |
1,2,7 |
10 |
Deficient |
| 15 |
1,3,5 |
9 |
Deficient |
| 16 |
1,2,4,8 |
15 |
Deficient |
| 17 |
1 |
1 |
Deficient |
| 18 |
1,2,3,6,9 |
21 |
Abundant |
| 19 |
1 |
1 |
Deficient |
| 20 |
1,2,4,5,10 |
22 |
Abundant |
There are many other theorems, conjectures, and postulates in number theory. A few of them are listed below.
- Chinese Remainder Theorem
- Fermat’s Little Theorem
- Pell Equations
- Euler Bricks
- Perfect Cuboids
- Sums of polygons
- Beal’s Conjecture
- Catalan’s conjecture
- Waring’s Problem
- Goldbach’s conjecture
- Weak Goldbach’s conjecture
- Fermat’s two square theorem
- Bertrand’s Postulate
- Andrica’s conjecture
- Legendre’s conjecture
- Twin prime conjecture
- De Polignac’s conjecture
- Chen’s theorems 1,2, and 3 variations on some other conjectures
- And many many more...