In mathematics, the Sieve of Eratosthenes is a simple and ingenious ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite the multiples of each prime, starting with the first prime number, 2; the multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them, equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime; the earliest known reference to the sieve is in Nicomachus of Gerasa's Introduction to Arithmetic, which describes it and attributes it to Eratosthenes of Cyrene, a Greek mathematician. One of a number of prime number sieves, it is one of the most efficient ways to find all of the smaller primes, it may be used to find primes in arithmetic progressions. A prime number is a natural number that has two distinct natural number divisors: the number 1 and itself.
To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n:. Let p equal 2, the smallest prime number. Enumerate the multiples of p by counting in increments of p from 2p to n, mark them in the list. Find the first number greater than p in the list, not marked. If there was no such number, stop. Otherwise, let p now equal this new number, repeat from step 3; when the algorithm terminates, the numbers remaining not marked in the list are all the primes below n. The main idea here is that every value given to p will be prime, because if it were composite it would be marked as a multiple of some other, smaller prime. Note that some of the numbers may be marked more than once; as a refinement, it is sufficient to mark the numbers in step 3 starting from p2, as all the smaller multiples of p will have been marked at that point. This means that the algorithm is allowed to terminate in step 4 when p2 is greater than n.
Another refinement is to list odd numbers only, count in increments of 2p from p2 in step 3, thus marking only odd multiples of p. This appears in the original algorithm; this can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few primes and not just from odds, counting in the correspondingly adjusted increments so that only such multiples of p are generated that are coprime with those small primes, in the first place. To find all the prime numbers less than or equal to 30, proceed as follows. First, generate a list of integers from 2 to 30: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 The first number in the list is 2; the numbers not crossed out at this point in the list are all the prime numbers below 30: 2 3 5 7 11 13 17 19 23 29 The sieve of Eratosthenes can be expressed in pseudocode, as follows: algorithm Sieve of Eratosthenes is input: an integer n > 1. Output: all prime numbers from 2 through n. let A be an array of Boolean values, indexed by integers 2 to n all set to true.
For i = 2, 3, 4... not exceeding √n do if A is true for j = i2, i2+i, i2+2i, i2+3i... not exceeding n do A:= false return all i such that A is true. This algorithm produces all primes not greater than n, it includes a common optimization, to start enumerating the multiples of each prime i from i2. The time complexity of this algorithm is O, provided the array update is an O operation, as is the case; as Sorenson notes, the problem with the sieve of Eratosthenes is not the number of operations it performs but rather its memory requirements. For large n, the range of primes may not fit in memory; the algorithm walks through the entire array A, exhibiting no locality of reference. A solution to these problems is offered by segmented sieves, where only portions of the range are sieved at a time; these have been known since the 1970s, work as follows: Divide the range 2 through n into segments of some size Δ ≤ √n. Find the primes in the first segment, using the regular sieve. For each of the following segments, in increasing order, with m being the segment's topmost value, find the primes in it as follows: Set up a Boolean array of size Δ, Eliminate from it the multip
Diamond Village is a small farming village on the windward side of St. Vincent, the main island of the archipelago nation of Saint Vincent and the Grenadines. Diamond Village referred to as "Diamonds," is located in the Parish of Charlotte and is part of the South Central Windward Parliamentary Constituency. Located 18 miles north-east of Kingstown, the capital of St. Vincent and the Grenadines, it takes about an hour to drive to or from Kingstown along the winding Windward Highway. Diamond Village proper has a population of 500, although when grouped with small surrounding villages such as New Adelphi and Stinking Tree, the population increases to about 900. There is a primary school in the village, Diamond Government School, a district clinic, a post office, a small community center, two churches, about 8 small shops; the majority of Diamond Village inhabitants are engaged in agriculture banana farming. However, as the profitability of bananas declines and the level of education increases, more young people are employed in the civil service in Kingstown.
Diamonites, the active youth group in the village, produced a documentary recounting the village's history through the oral histories of village elders. The documentary is entitled Connecting Generations: A History of Diamond Village. Coordinates are estimated from a map provided by the St Vincent and the Grenadines Ministry of Tourism and Culture
Mangai Oru Gangai is a 1987 Indian Tamil-language legal drama film directed by T. Hariharan and produced by Kovaithambi; the film stars Nadhiya in the lead roles. It was released on 24 July 1987. Radha Ranganathan, an advocate, becomes cynical after failing to get a criminal convicted for the murder of her brother-in-law. Saritha as Radha Ranganathan Nadhiya as Radha's protege Charan Raj Poornam Vishwanathan Suresh Mangai Oru Gangai was directed by T. Hariharan and produced by Kovaithambi of Motherland Pictures. Hariharan wrote the screenplay, based on a story by Motherland Pictures' Story Department, the dialogue was written by M. G. Vallabhan. Cinematography was handled by U. Rajagopal, editing by M. S. Money; the soundtrack was composed by the duo Laxmikant–Pyarelal. Mangai Oru Gangai was released on 24 July 1987. On 7 August, N. Krishnaswamy of The Indian Express wrote, "I was pleased that the makers of the film had done their homework on the script and had here something, intellectually simulating", praised the performances of Saritha, Charan Raj and Vishwanathan.
Mangai Oru Gangai on IMDb