Mathematics 4 You

I want to know God's thoughts; the rest are details. ~Albert Einstein

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 a2+b2=c2 has many solutions, (32+42=52 for example) a3+b3=c3 has no integer solutions, a4+b4=c4 has no integer solutions, a5+b5=c5 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: 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.