Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • GIMPS sets new prime record

      2 January, 2019

      Number lovers around the world received a special treat this Christmas: on December 21, 2018, the discovery of a previously unknown prime number was announced.

      282,589,933−1 has an impressive 24,862,048 digits, which makes it the largest prime number known to humanity.
      New prime

      Some of the nearly 25 million digits of the newly discovered prime.

      Primes are natural numbers that can only be divided by 1 and themselves. The first five primes are 2, 3, 5, 7, and 11. People have known since the time of the ancient Greeks that there are infinitely many prime numbers, but there is no quick and easy way to find them all. This is why every new discovery of a prime number makes mathematical headlines.

      The latest addition to the list of known primes was discovered by the Great Internet Mersenne Prime Search (GIMPS), a grassroots computing project in which volunteers download a free program which searches for prime numbers and run it on their computers. Everyone lucky enough to find a new prime gets a cash reward.

      This time around the lucky discoverer was Patrick Laroche from Ocala, Florida, who has been using GIMPS software to stress test his computer builds for years. He only started actually hunting for primes recently and was rewarded after less than four months on only his fourth try. It's an extremely lucky find: some GIMPS prime hunters have searched for over twenty years without success. Laroche is eligible for $3,000 in cash as a reward.

      In its search for new primes GIMPS only considers numbers that are of the form 2n−1, where n is a known prime (for the new discovery, n=82,589,933). It's quicker to test whether these Mersenne numbers, as they are called, are prime then it is to test other numbers. But this doesn't mean it's an easy feat: the proof that the latest discovery really is prime took twelve days of non-stop computing on a machine with an Intel i5-4590T CPU.

      Laroche's lucky discovery comes within a lucky streak in the GIMPS project as a whole, which may shed light on how the Mersenne primes are distributed among the other numbers. By looking at the distribution of the first Mersenne primes to be discovered mathematicians had come up with a formula for working out, at least asymptotically, the number of Mersenne primes you might expect to find in a given stretch of the number line. Yet, the number of Mersenne primes that GIMPS has found in the stretch from

      220,000,000−1to285,000,000−1.

      is twelve: that's three times as many as the formula would suggest. "This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes [are] incorrect," says the GIMPS website. "However, if the trend continues it may be worth further investigation."

      Mersenne primes are interesting for another reason too. Since the time of the ancient Greeks, and perhaps before, people have been beguiled by perfect numbers: natural numbers which are equal to the sum of their proper devisors. The smallest perfect number is 6 because 6=1×2×3=1+2+3. The next perfect number along is 28, followed by 496 and 8128.

      Perfect numbers are rare gems — the ancients only knew the four mentioned above. However, Euclid proved that once you get your hands on a Mersenne prime you get an even perfect number for free: if the Mersenne prime is 2n−1, then the number 2n−1(2n−1) is perfect. In the 18th century Leonhard Euler proved that every even perfect number arises from a Mersenne prime in this way.

      The perfect number stemming from the newly discovered Mersenne prime is 282,589,932(282,589,933−1) and has over 49 million digits. We now know of 51 perfect numbers and 51 Mersenne primes.

      Will we continue to find new Mersenne primes and perfect numbers? We can't be sure — mathematicians believe that there are infinitely many of both, so the supply will never run out, but nobody has been able to prove this. We also don't know if all perfect numbers are even, as the ones discovered so far, or whether there also are odd ones. These are some of the great mysteries of mathematics.

      Read more about...
      prime number search
      prime number
      Mersenne prime
      GIMPS
      Mersenne search
      perfect number
      number theory
      • Log in or register to post comments

      Comments

      Tony Reix

      22 January 2019

      Permalink

      Interesting news. I missed it. Some years ago I've contributed to checking that Mersenne Prime candidates are true primes. And I learnt a lot about how it is done. At least about the theoretical stuff, not about how the computation is done with a FFT. And this "theoretical" stuff deals with the proof that enables to say if a Mersenne number is prime or not, which is called: LLT : Lucas-Lehmer Test, based on the name of the two guys who provided fundamental work. Lucas built the idea and he provided not-perfect proofs. And then later Lehmer provided complete and clear proof. How this LLT proof works is astonishing. Proving it may be very complex, but understanding how it works is easy and funny. I think that you should talk about this LLT mechanism here. I could help if needed. Contact me at: tony(.)reix[@]laposte(.)net . Regards. Grenoble. France

      • Log in or register to post comments

      Adams

      23 January 2019

      Permalink

      Lets see the Secrets of Prime Numbers when I divide 1 to prime numbers I see the same occurence for example 1/13. Our number is 0.076923.

      76923 x 1 = 076 923

      76923 x 2 = 153 846

      76923 x 3 = 230 769

      76923 x 4 = 307 692

      76923 x 5 = 384 615

      76923 x 6 = 461 538

      76923 x 7 = 538 461

      76923 x 8 = 615384

      76923 x 9 = 692307

      76923 x 10 = 769230

      76923 x 11 = 846153

      76923 x 12 = 923076

      76923 x 13 = 999999

      can you see the iteration of same numbers 076923 and did you notice the 076+923=999

      ***So why this iterates ???***

      Besides, I did it for other small or very large prime numbers and come across more strange results :) Lets look..

      https://googleofscience.wordpress.com/secrets-of-prime-numbers/

      • Log in or register to post comments

      Read more about...

      prime number search
      prime number
      Mersenne prime
      GIMPS
      Mersenne search
      perfect number
      number theory
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms