Sieve of Sundaram

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the sieve of Sundaram is a simple deterministic algorithm for finding all prime numbers up to a specified integer. It was discovered in 1934 by S. P. Sundaram, an Indian student from Sathyamangalam.[1][2]

Algorithm[edit]

Sieve of Sundaram: algorithm steps for primes below 202 (unoptimized).

Start with a list of the integers from 1 to n. From this list, remove all numbers of the form i + j + 2ij where:

  • i,j\in\mathbb{N},\ 1 \le i \le j
  • i + j + 2ij \le n

The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (i.e., all primes except 2) below 2n + 2.

The sieve of Sundaram sieves out the composite numbers just as sieve of Eratosthenes does, but even numbers are not considered; the work of "crossing out" the multiples of 2 is done by the final double-and-increment step. Whenever Eratosthenes' method would cross out k different multiples of a prime 2i+1, Sundaram's method crosses out i + j(2i+1) for 1\le j\le \lfloor k/2\rfloor.

Correctness[edit]

The final list of doubled-and-incremented integers contains only odd integers; we must show that the set of odd integers excluded from the list is exactly the set of composite odd integers.

An odd integer is excluded from the final list if and only if it is of the form 2(i + j + 2ij) + 1, and we have

2(i + j + 2ij) + 1
= 2i + 2j + 4ij + 1
= (2i + 1)(2j + 1).

So, an odd integer is excluded from the final list if and only if it has a factorization of the form (2i + 1)(2j + 1) — which is to say, if it has a non-trivial odd factor. Therefore the list must be composed of exactly the set of odd prime numbers less than or equal to 2n + 2.

See also[edit]

References[edit]

  1. ^ V. Ramaswami Aiyar (1934). "Sundaram's Sieve for Prime Numbers". The Mathematics Student 2 (2): 73. ISSN 0025-5742. 
  2. ^ G. (1941). "Curiosa 81. A New Sieve for Prime Numbers". Scripta Mathematica 8 (3): 164.