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The following table lists the progression of the largest known prime number in ascending order. [3] Here M p = 2 p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M 19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456. [citation needed]
Of these, the greatest is 6, so it is the greatest common divisor: (,) = Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient methods are described in § Calculation.
The number of steps to calculate the GCD of two natural numbers, a and b, may be denoted by T(a, b). [96] If g is the GCD of a and b, then a = mg and b = ng for two coprime numbers m and n. Then T(a, b) = T(m, n) as may be seen by dividing all the steps in the Euclidean algorithm by g. [97]
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. gcd( m , n ) ( greatest common divisor of m and n ) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n ).
Euler ascertained that 2 31 − 1 = 2147483647 is a prime number; and this is the greatest at present known to be such, and, consequently, the last of the above perfect numbers [i.e., 2 30 (2 31 − 1)], which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for as they ...
Pascal's calculator (also known as the arithmetic machine or Pascaline) is a mechanical calculator invented by Blaise Pascal in 1642. Pascal was led to develop a calculator by the laborious arithmetical calculations required by his father's work as the supervisor of taxes in Rouen . [ 2 ]
Note: Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent. 10080 is a so-called 7-smooth number (sequence A002473 in the OEIS ) .