mathematics

List of largest known primes and probable primes

List of largest known primes and probable primes is a mathematics 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 List of largest known primes and probable primes rather than just read about it. In short: The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.

Key takeaways

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

Reference excerpt

The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.

Largest known primes These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and Φ 3 ( x ) {\displaystyle \Phi _{3}(x)} is the third cyclotomic polynomial, defined as x 2 + x + 1 {\displaystyle x^{2}+x+1} .

Largest known probable primes (PRPs) These are probable primes. Primality has not been proven because it is too hard for general numbers of this size but they are expected to be primes. F(n) is the nth Fibonacci number.

Worked examples

Example 1 — a first encounter with List of largest known primes and probable primes

Start with the simplest possible case. Write down what List of largest known primes and probable primes claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 List of largest known primes and probable primes 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 List of largest known primes and probable primes 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 List of largest known primes and probable primes

In research: List of largest known primes and probable primes appears in mathematics 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 List of largest known primes and probable primes 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: List of largest known primes and probable primes is common in secondary-school and first-year university syllabi. It links to neighbouring topics Large integers, Prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for List of largest known primes and probable primes 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “List of largest known primes and probable primes” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study List of largest known primes and probable primes in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what List of largest known primes and probable primes 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 List of largest known primes and probable primes out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is List of largest known primes and probable primes in simple terms?

Why does List of largest known primes and probable primes matter?

Because it connects several mathematics 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 List of largest known primes and probable primes?

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 List of largest known primes and probable primes.