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Largest known prime number

Largest known prime number 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 Largest known prime number rather than just read about it. In short: The largest known prime number as of August 2026 is 2136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).

Largest known prime number — main illustration
Largest known prime number — illustration

Key takeaways

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

Reference excerpt

The largest known prime number as of August 2026 is 2136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).

A prime number is a natural number greater than 1 with no divisors other than 1 and itself. Euclid's theorem proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; there is no largest prime. Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can be verified by a specialized primality test that is faster than the general one. As of October 2024, the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones. Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not true. While large primes with hundreds of digits are indeed used for cryptography, primes with millions of digits are not.

Current record

The record is held by 2136,279,841 − 1 with 41,024,320 digits, found by GIMPS on October 12, 2024. Its first and last 120 digits are:

881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874... [41,024,080 digits skipped] ...852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

Prizes The Electronic Frontier Foundation (EFF) offers several prizes for record primes. A prime with one million digits was found in 1999, earning the discoverer a $50,000 prize. In 2008, a ten-million-digit prime won a $100,000 prize and a Cooperative Computing Award from the EFF. Time called this prime the 29th top invention of 2008. Both these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the $150,000 prize for the first prime of over 100 million digits with the winning participant. A $250,000 prize awaits the first prime with one billion digits. GIMPS also offers a $3,000 research discovery award for participants who discover a new Mersenne prime of fewer than 100 million digits.

History

The following table lists the progression of the largest known prime number in ascending order. Here Mp = 2p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder was M19 = 524,287, which held the record for 184 years. The primes up to and including 2 148 + 1 17 {\displaystyle {\tfrac {2^{148}+1}{17}}} were found without a computer, while the primes starting with 180×(M127)2+1 were found using computers. GIMPS volunteers found the 16 latest records, all of them Mersenne primes. They were found on ordinary personal computers until the most recent one, found by ex-Nvidia employee Luke Durant using a network of thousands of dedicated graphics processing units (GPUs). Durant spent about a year and $2 million on the hunt. This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).

Twenty largest A list of the 5,000 largest known primes is maintained by the PrimePages, of which the largest 20 are listed below.

See also List of largest known primes and probable primes

References

External links Press release about the former largest known prime 274,207,281−1 Press release about the former largest known prime 277,232,917−1 Press release about the former largest known prime 282,589,933−1 Press release about the largest known prime 2136,279,841−1

Illustrations

Largest known prime number: A plot of the number of digits in the largest known prime by year since the advent of the electronic computer. The vertical scale is logarithmic.
A plot of the number of digits in the largest known prime by year since the advent of the electronic computer. The vertical scale is logarithmic.
Largest known prime number: Using standard A4 paper format of 50 lines per page and 75 characters per line, it would require 10,940 single-sided pages to print this prime number, or approximately 22 reams of paper. (15 are shown here.)[9]
Using standard A4 paper format of 50 lines per page and 75 characters per line, it would require 10,940 single-sided pages to print this prime number, or approximately 22 reams of paper. (15 are shown here.)[9]
Largest known prime number: Commemorative postmark used by the UIUC Math Department after proving that M11213 is prime
Commemorative postmark used by the UIUC Math Department after proving that M11213 is prime

Worked examples

Example 1 — a first encounter with Largest known prime number

Start with the simplest possible case. Write down what Largest known prime number 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 Largest known prime number 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 Largest known prime number 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 Largest known prime number

In research
Largest known prime number 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 Largest known prime number 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
Largest known prime number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Great Internet Mersenne Prime Search, Large integers, Largest things, so understanding it makes those chapters shorter.
In everyday life
Look for Largest known prime number 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 Largest known prime number in 20 minutes

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

Frequently asked questions

What is Largest known prime number in simple terms?

The largest known prime number as of August 2026 is 2136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a researcher from San Jose, California, to the Great Internet…

Why does Largest known prime number 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 Largest known prime number?

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 Largest known prime number.

Tags

  • Great Internet Mersenne Prime Search
  • Large integers
  • Largest things
  • Mersenne primes
  • Prime numbers
  • World records

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