Factorials Hard

Factorials: Definition, History, Applications, and Properties

• The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.

• Factorials have been discovered in several ancient cultures, including Indian mathematics and Jewish mysticism.

• Factorials are used in combinatorics to count the possible distinct sequences of n distinct objects.

• Factorials are also used in power series for the exponential function and other functions, and have applications in algebra, number theory, probability theory, and computer science.

• The growth rate of factorials is faster than exponential growth but slower than a double exponential function.

• Stirling's approximation provides an accurate approximation to the factorial of large numbers.

• Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials.

• The factorial function can be extended to a continuous function of complex numbers, except at the negative integers, the gamma function.

• Other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials.

• Implementations of the factorial function are commonly used as an example of different computer programming styles.

• Factorials have applications in probability theory, quantum and statistical physics, and computer science.

• The product formula for the factorial implies that n! is divisible by all prime numbers that are at most n, and by no larger prime numbers.The Factorial Function: Definition, Properties, and Computation

• The factorial function is a mathematical function that takes a non-negative integer as input and returns the product of all positive integers up to and including that integer.

• The factorial function is denoted by an exclamation mark (!) after the input integer, e.g., n! denotes the factorial of n.

• The factorial function grows very quickly, as n! = 1 x 2 x 3 x ... x n, and its value for large n can be approximated by Stirling's formula.

• The factorial function has many applications in combinatorics, probability theory, and number theory, such as counting permutations, combinations, and partitions.

• The factorial function satisfies a recurrence relation, n! = n x (n-1)!, and has a closed-form expression in terms of gamma function, (n+1)! = Γ(n+2).

• The factorial function has many properties, such as being an integer, being divisible by all integers up to n, and being related to the binomial coefficients.

• The factorial function can be extended to non-integer values using the gamma function, which is a continuous interpolation of the factorials.

• The gamma function satisfies a functional equation, Γ(z+1) = z Γ(z), and has a unique log-convexity property that distinguishes it from other continuous interpolations.

• The factorial function can be computed in various ways, such as by iteration, recursion, memoization, dynamic programming, and prime factorization.

• The computational complexity of the factorial function depends on the method used, and can range from O(n) to O(n log^2 n) in time and O(1) to O(n log n) in space.

• The factorial function can be used to define and compute other related sequences and functions, such as the harmonic numbers, the digamma function, and the p-adic gamma function.

• The factorial function is a common feature in scientific calculators and programming libraries, and its efficient computation is important in many scientific and engineering applications.

Factorials: Definition, History, Applications, and Properties

• The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.

• Factorials have been discovered in several ancient cultures, including Indian mathematics and Jewish mysticism.

• Factorials are used in combinatorics to count the possible distinct sequences of n distinct objects.

• Factorials are also used in power series for the exponential function and other functions, and have applications in algebra, number theory, probability theory, and computer science.

• The growth rate of factorials is faster than exponential growth but slower than a double exponential function.

• Stirling's approximation provides an accurate approximation to the factorial of large numbers.

• Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials.

• The factorial function can be extended to a continuous function of complex numbers, except at the negative integers, the gamma function.

• Other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials.

• Implementations of the factorial function are commonly used as an example of different computer programming styles.

• Factorials have applications in probability theory, quantum and statistical physics, and computer science.

• The product formula for the factorial implies that n! is divisible by all prime numbers that are at most n, and by no larger prime numbers.The Factorial Function: Definition, Properties, and Computation

• The factorial function is a mathematical function that takes a non-negative integer as input and returns the product of all positive integers up to and including that integer.

• The factorial function is denoted by an exclamation mark (!) after the input integer, e.g., n! denotes the factorial of n.

• The factorial function grows very quickly, as n! = 1 x 2 x 3 x ... x n, and its value for large n can be approximated by Stirling's formula.

• The factorial function has many applications in combinatorics, probability theory, and number theory, such as counting permutations, combinations, and partitions.

• The factorial function satisfies a recurrence relation, n! = n x (n-1)!, and has a closed-form expression in terms of gamma function, (n+1)! = Γ(n+2).

• The factorial function has many properties, such as being an integer, being divisible by all integers up to n, and being related to the binomial coefficients.

• The factorial function can be extended to non-integer values using the gamma function, which is a continuous interpolation of the factorials.

• The gamma function satisfies a functional equation, Γ(z+1) = z Γ(z), and has a unique log-convexity property that distinguishes it from other continuous interpolations.

• The factorial function can be computed in various ways, such as by iteration, recursion, memoization, dynamic programming, and prime factorization.

• The computational complexity of the factorial function depends on the method used, and can range from O(n) to O(n log^2 n) in time and O(1) to O(n log n) in space.

• The factorial function can be used to define and compute other related sequences and functions, such as the harmonic numbers, the digamma function, and the p-adic gamma function.

• The factorial function is a common feature in scientific calculators and programming libraries, and its efficient computation is important in many scientific and engineering applications.