Number Theory:
Number theory is a branch of mathematics concerned with the properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural of mathematical pursuits.
Number theory is the study of the integers (i.e. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis.
Until the mid-20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. The advent of digital computers and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes, testing conjectures, and solving numerical problems once considered out of reach.
Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers.
We will study those topics on number theory for this class;
- Integers
- Modular Arithmetic
- Diophantine Equations
Remainder:
In integer division, the remainder is the amount that is left over after dividing one integer by another. In polynomial division, the remainder is the polynomial that is left over after dividing one polynomial by another.
Integer Division:
An integer “a” is divisible by another integer “b” (or is a multiple of “b” ) it can be written as times another integer;
a = b x (integer)
Note that the remainder (r) will always be less than the divisor (d).
x=xd+ r where r <d
Primes:
A prime number is a positive integer that has exactly 2 positive divisors. i.e. 1 and that respective number only. The first few prime numbers are;
2,3,5,7,11,13,17……………….
Euclid’s Theorem: There are infinitely many primes.
Euclid’s Proof;
For any finite set of primes {, , Euclid considered the number
“n” has a prime divisor “p” (every integer has at least one prime divisor). But “p” is not equal to any of the (If “p” were equal to any of the , then “p” would have to divide 1 , which is impossible.) So for any finite set of prime numbers, it is possible to find another prime that is not in that set.
In other words, a finite set of primes cannot be the collection of all prime numbers.
Notice that Euclid’s original proof was direct proof, not proof by contradiction.
Largest known prime number:
Despite there being infinitely many prime numbers, it’s actually difficult to find a large one. For recreational purposes, people have been trying to find as a large prime number as possible. The current largest known prime number is, having, 22,338,618 digits. This was found by the Great Internet Mersenne Prime Search project, which uses distributed computing to discover prime numbers in the form, known as Mersenne primes. Mersenne primes are very rare; in fact, it’s an open problem whether there are infinitely many Mersenne primes or not. Only 49 are known so far, the largest of which is the above.
Prime Factorization:
In number theory, the prime factorization of a number N is the set consisting of prime numbers whose product is N. As an example, the prime factorization of 90 is
90 = 2 X 3 X 3 X 5.
Due to its uniqueness for every positive integer, the prime factorization provides a foundation for elementary number theory.
Prime Factors:
The uniqueness of prime factorization is an incredibly important result, thus earning the name of the fundamental theorem of arithmetic:
Fundamental theorem of arithmetic: Any integer greater than 1 is either a prime number or can be written as a unique product of prime numbers.
This statement implies that if a number is not prime, it has a prime number as its factor. For example, the factors of 8 are 1, 2, 4, and 8, where 2 and 4 are both prime numbers.
Example:
If x, y, z are three different prime numbers such that N =, how many positive divisors does N have excluding 1 and itself?
Since N =, we can conclude that x, y, and z are the factors of N . Since x, y and z are prime numbers, we can’t factor them to get any other number, so that gives us a total of 3 numbers.
But wait, we know that if x and y are factors of N, then y is also a factor of N. So a combination of two factors out of the three factors is also a divisor of N. In other words, we have, and as factors of N which are another 3 in addition to the 3 above.
Note that is also a combination that is a factor of N, but it equals the number itself and is therefore omitted.
So we have a total of 6 divisors, excluding 1 and the number itself.
Perfect Squares, Cubes, and Powers:
A perfect square is an integer that can be expressed as the product of two equal integers. For example, 100 is a perfect square because it is equal to. If N is any integer, then is a perfect square. Because of this definition, perfect squares are always non-negative.
Similarly, a perfect cube is an integer that can be expressed as the product of three equal integers. For example, 27 is a perfect cube because it is equal to 3. Determining if a number is a perfect square, cube, or higher power can be determined from the prime factorization of the number.
Perfect Numbers: A perfect number is a positive integer that equals the sum of its proper divisors, that is, positive divisors excluding the number itself. For example, 6 is a perfect number because the proper divisors of 6 are 1, 2 and 3 and 6
The sum of all positive divisors of a number n is denoted by. A perfect number is therefore a positive integer n such that
Perfect numbers were of great interest to ancient mathematicians including the Greeks, who attributed mystical significance to the property. It is perhaps somewhat surprising that many elementary questions about perfect numbers are still open. Most notably, it is not known whether there are infinitely many perfect numbers, and it is not known whether there are any odd perfect numbers.
Theorem: A positive integer “n” is an even perfect number if and only if for some positive prime “p” such that is prime. Here is known as “Mersenne Prime”
Properties of Perfect Number:
- Every even perfect number ends in the digits 6 or 28 (when written in decimal).
- The iterative digital sum of any even perfect number except 6 is 1.
- The only square-free perfect number is 6.
- A perfect number is an Ore harmonic number; that is, the harmonic mean of its divisors is an integer. (Not every Ore harmonic number is perfect, e.g. 140.)
- If an odd perfect number exists, it has more than 300 digits, at least 75 prime factors, at least 9 distinct prime factors and at least one prime factor of at least 20 digits.
Number Base:
A number base (or base for short) of a numeral system tells us about the unique or different symbols and notations it uses to represent a value.
For example, the number base 2 tells us that there are only two unique notations 0 and 1.
The most common number base is decimal, also known as base 10. The decimal number system uses 10 different notations which are the digits 0~9. The base is not necessarily positive integers. The base can be positive, negative and 0, complex and nonintegral. Other frequently used bases include bases 2 and 16. These are used in computing and are called binary and hexadecimal.
****** I have posted this blog in hurry, Will make corrections soon. Especially the character and equation are missing from the above post. I’ll make corrections to those. Thanks
