space requirements to the very minimal size of the segment page plus the memory required to store the base primes less than the square root of the range used to cull composites from successive page segments of size . A special (rarely, if ever, implemented) segmented version of the sieve of Eratosthenes, with basic optimizations, uses operations and bits of memory. Using big O notation ignores constant factors and offsets that may be very significant for practical ranges: The sieve of Eratosthenes variation known as the Pritchard wheel sieve has an performance, but its basic implementation requires either a "one large array" algorithm which limits its usable range to the amount of available memory else it needs to be page segmented to reduce memory use. When implemented with page segmentation in order to save memory, the basic algorithm still requires about bits of memory (much more than the requirement of the basic page segmented sieve of Eratosthenes using bits of memory). Pritchard's work reduced the memory requirement at the cost of a large constant factor. Although the resulting wheel sieve has performance and an acceptable memory requirement, it is not faster than a reasonably Wheel Factorized basic sieve of Eratosthenes for practical sieving ranges. Euler's sieve Euler's proof of the zeta product formula contains a version of the sieve of Eratosthenes in which each composite number is eliminated exactly once. The same sieve was rediscovered and observed to take linear time by . It, too, starts with a list of numbers from 2 to in order. On each step the first element is identified as the next prime, is multiplied with each element of the list (thus starting with itself), and the results are marked in the list for subsequent deletion. The initial element and the marked elements are then removed from the working sequence, and the process is repeated: [2] (3) 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 ... [3] (5) 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 53 55 59 61 65 67 71 73 77 79 ... [4] (7) 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 71 73 77