science

Legendre sieve

Legendre sieve is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Legendre sieve rather than just read about it. In short: In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers.

Key takeaways

Legendre sieve belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Legendre sieve to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Legendre sieve from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers. Because it is a simple extension of Eratosthenes' idea, it is sometimes called the Legendre–Eratosthenes sieve.

Legendre's identity The central idea of the method is expressed by the following identity, sometimes called the Legendre identity:

S ( A , P ) = ∑ a ∈ A ∑ d ∣ a ; d ∣ P μ ( d ) = ∑ d ∣ P μ ( d ) | A d | , {\displaystyle S(A,P)=\sum _{a\in A}\sum _{d\mid a;\,d\mid P}\mu (d)=\sum _{d\mid P}\mu (d)|A_{d}|,}

where A is a set of integers, P is a product of distinct primes, μ {\displaystyle \mu } is the Möbius function, and A d {\displaystyle A_{d}} is the set of integers in A divisible by d, and S(A, P) is defined to be:

S ( A , P ) = | { n : n ∈ A , gcd ( n , P ) = 1 } | {\displaystyle S(A,P)=|\{n:n\in A,\gcd(n,P)=1\}|}

i.e. S(A, P) is the count of numbers in A with no factors common with P. Note that in the most typical case, A is all integers less than or equal to some real number X, P is the product of all primes less than or equal to some integer z < X, and then the Legendre identity becomes:

S ( A , P ) =

∑ d ∣ P μ ( d ) ⌊ X d ⌋ =

⌊ X ⌋ − ∑ p 1 ≤ z ⌊ X p 1 ⌋ + ∑ p 1 < p 2 ≤ z ⌊ X p 1 p 2 ⌋

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Legendre sieve

Start with the simplest possible case. Write down what Legendre sieve claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Legendre sieve before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Legendre sieve ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Legendre sieve

In research: Legendre sieve appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Legendre sieve in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Legendre sieve is common in secondary-school and first-year university syllabi. It links to neighbouring topics Sieve theory, so understanding it makes those chapters shorter.
In everyday life: Look for Legendre sieve outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Legendre sieve in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Legendre sieve means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Legendre sieve out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Legendre sieve in simple terms?

Why does Legendre sieve matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Legendre sieve?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Legendre sieve.