Problems
Unfortunately, the Legendre sieve has a problem with fractional parts of terms accumulating into a large error, which means the sieve only gives very weak bounds in most cases. For this reason it is almost never used in practice, having been superseded by other techniques such as the Brun sieve and Selberg sieve. However, since these more powerful sieves are extensions of the basic ideas of the Legendre sieve, it is useful to first understand how this sieve works.
Read more about this topic: Legendre Sieve
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