Number Theory

 Number Theory & Programmings

Basically we all are know about different numbers in mathamatics like natural numbers whole numbers etc....

But in some advanced mathamatical modelling we need and we determine some special number cases these have unique properties and used in different fields like computers, quantum mechanics, cosmology, and other ratiinal fields.

Fist we started with basic numbers and finally discuss special numbers

• Natural Numbers: The counting numbers 1, 2, 3, 4, 5, and so on.
• Whole Numbers: The numbers 0, 1, 2, 3, 4, and so on.
• Integers: The numbers ...,-4,-3,-2,-1,0,1,2,3,4,.... (positive and negative whole numbers, including zero)
• Rational Numbers: Numbers that can be expressed as a ratio of two integers (fractions).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction of two integers and have an infinite, non-repeating decimal representation. Examples include √2, π (pi).
• Real Numbers: The set of all rational and irrational numbers, including positive and negative numbers.
• Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (equal to the square root of -1).
• Odd Numbers: Integers that are not divisible by 2 (1, 3, 5, 7, 9, etc.).
• Even Numbers: Integers that are divisible by 2 (0, 2, 4, 6, 8, etc.

Special Cases:

Varieties in prime numbers:

1.Composite Numbers: Natural numbers greater than 1 that are not prime and have multiple factors.

2.Prime Numbers: Natural numbers greater than 1 that are only divisible by 1 and themselves.

3.Irregular prime: An irregular prime is a prime number that is greater than the average gap between consecutive primes.

4.Pseudoprime numbers: A pseudoprime number is a composite number that appears to be prime when tested using certain prime tests, but is not actually prime.

5.Sophie Germain primes: A Sophie Germain prime is a prime number that is also a prime when multiplied by 2 and added to 1.

6.Super prime: A super prime is a prime number that is also a prime when concatenated with other prime numbers.

7.Additive primes: An additive prime is a prime number that remains prime when a certain digit is added to it.

8.Permutable primes: A permutable prime is a prime number that remains prime when its digits are rearranged.

Sequences:

Pythagorean triples: A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem (a^2 + b^2 = c^2).

Euler's Totient Function: The Euler's Totient Function is a function that gives the number of positive integers less than a given integer that are relatively prime to it.

Fibonacci sequence: The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers, starting with 0 and 1 (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.).

Lucas sequence: The Lucas sequence is a series of numbers similar to the Fibonacci sequence, but with a different starting point (2, 1, 3, 4, 7, 11, etc.).

Catalan numbers: The Catalan numbers are a sequence of natural numbers that occur in various counting problems in combinatorics.

Bernoulli numbers: The Bernoulli numbers are a sequence of rational numbers that occur in various mathematical problems, such as the calculation of the Taylor series expansion of a certain function.

Bell numbers: The Bell numbers are a sequence of natural numbers that occur in various counting problems in combinatorics.

Harmonic numbers: The Harmonic numbers are a sequence of numbers that arise in the calculation of various mathematical series.

Stirling numbers: The Stirling numbers are a sequence of numbers that arise in various counting problems in combinatorics and probability.

Motzkin numbers: The Motzkin numbers are a sequence of numbers that arise in various counting problems in combinatorics and algebra.

Tribonacci sequence: The Tribonacci sequence is a series of numbers in which each number is the sum of the three preceding numbers, starting with 0, 0, and 1 (0, 0, 1, 1, 2, 4, 7, 13, 24, 44, etc.).

Jacobsthal numbers: The Jacobsthal numbers are a sequence of natural numbers that arise in various mathematical problems, such as the calculation of the number of ways to tile a strip with certain types of tiles.

Recaman's sequence: Recaman's sequence is a series of numbers in which each number is the difference between the previous number and its index in the series.

1.Palindrome Numbers: Numbers that remain the same when their digits are reversed (e.g. 121, 787).

2.Strong Numbers: A positive integer is called a strong number if the sum of the factorial of its digits is equal to the number itself (e.g. 145).

3.Armstrong Numbers: A number is called an Armstrong number if the sum of its digits, each raised to the power of the number of digits, is equal to the number itself (e.g. 153, 371).

4.Perfect Numbers: A positive integer is called a perfect number if the sum of its proper divisors (excluding itself) is equal to the number itself (e.g. 6, 28).

5.Deficient Numbers: A positive integer is called deficient if the sum of its proper divisors (excluding itself) is less than the number itself.

6.Abundant Numbers: A positive integer is called abundant if the sum of its proper divisors (excluding itself) is greater than the number itself.

7.narcissistic numbers: A number is called a narcissistic number if the sum of its digits, each raised to the power of the number of digits, is equal to the number itself (e.g. 153, 371).

8.triangular numbers: A triangular number is a number that can be represented as the sum of a series of consecutive natural numbers, starting from 1 (e.g. 6, which can be represented as 1 + 2 + 3).

9.square numbers: A square number is a number that can be represented as the result of squaring an integer (e.g. 4, which is equal to 2^2).

10.cube numbers: A cube number is a number that can be represented as the result of cubing an integer (e.g. 8, which is equal to 2^3).

11.Fibonacci numbers: The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones (e.g. 1, 1, 2, 3, 5, 8, 13, 21, etc.).

12.Catalan numbers: The Catalan numbers are a sequence of numbers that arise in combinatorial mathematics and are used in various counting problems.

13.Bell numbers: The Bell numbers are a sequence of numbers that count the number of ways to partition a set.

14.Mersenne primes: A Mersenne prime is a prime number of the form 2^p - 1, where p is also a prime number.

15.Happy numbers: A happy number is a number that eventually reaches 1 when you apply the sum of the squares of its digits in a repeating process.

16.Unhappy numbers: An unhappy number is a number that does not eventually reach 1 when you apply the sum of the squares of its digits in a repeating process.

17.Amicable numbers: Amicable numbers are pairs of numbers such that the sum of the proper divisors of one number is equal to the other number, and vice versa.

18.Automorphic numbers: A number is called automorphic if its square ends with the original number (e.g. 5^2 = 25).

19.Harshad numbers: A number is called a Harshad number if it is divisible by the sum of its digits (e.g. 18).

20.Padovan sequence: The Padovan sequence is a series of numbers in which each number is the sum of the two preceding numbers, with a different starting point from the Fibonacci sequence (e.g. 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, etc.).

21.Lucas numbers: The Lucas numbers are a series of numbers similar to the Fibonacci sequence, but with a different starting point (e.g. 2, 1, 3, 4, 7, 11, etc.).

22.Magic numbers: A magic number is a number that appears frequently in various counting and numbering systems, such as 7 in the number of days in a week, or 12 in the number of signs in the zodiac.

23.Fortunate numbers: A number is called fortunate if it appears frequently in various counting and numbering systems, or if it is considered to bring good luck in various cultures.

24.Semiperfect numbers: A number is called semiperfect if it is equal to the sum of some of its proper divisors (excluding itself).

25.Superabundant numbers: A number is called superabundant if the sum of its proper divisors (excluding itself) is greater than twice the number itself.

26.Strobogrammatic numbers: A number is called strobogrammatic if it reads the same when rotated 180 degrees (e.g. 69, 88, 96).

27.Lucky numbers: A lucky number is a number that is considered to bring good luck in various cultures.

28.Unlucky numbers: An unlucky number is a number that is considered to bring bad luck in various cultures.

29.Emirp numbers: An emirp number is a prime number that is also a prime when its digits are reversed.

30.Kaprekar numbers: A Kaprekar number is a number that, when squared, can be split into two parts that add up to the original number.

31.Cycle numbers: A cycle number is a number that, when multiplied by a certain integer, cycles through all the digits of the number.

32.Kaprekar constant: The Kaprekar constant is a number that arises in the study of Kaprekar numbers and has many interesting properties.