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919444¹⁰⁴⁸⁵⁷⁶ + 1 is prime

Distributed internet prime number search PrimeGrid has found a new largest generalised Fermat prime.

The discovery was made on 29th August, and was double-checked before being announced on 2nd September. PrimeGrid uses a distributed computing approach and uses spare computer time donated by volunteer computers connected to their network.

A generalised Fermat prime is a prime number of the form $a^{2^n} + 1$, with $a \gt 0$. It’s called ‘generalised’ because ‘Fermat prime’ is the name for the particular case $a=2$.

Much like Mersenne primes, there are special tests which make it much easier to check if a number of this form is prime than for a general number. For this reason, they’re a good place to look for new large primes.

Until now only 392 generalised Fermat primes had been found: this new discovery makes 393. At 6,253,210 digits long, it’s now the 12th largest of all known primes, and the second-largest known non-Mersenne prime.

PrimeGrid have put out an announcement in PDF format giving some more details about the search, and credits for the many people involved writing algorithms and providing computers to run them on.

The PrimeGrid homepage has more information about the many different prime number searches they run, and how to join in the search with your own PC.

New Twin Primes found

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Collaborative prime number searching website PrimeGrid has announced its most recent discovery: on 14th September, user Tom Greer discovered a new pair of twin primes (primes which differ by 2), namely:

\[2996863034895 \times 2^{1290000} \pm 1\]

Found using PrimeGrid’s Sophie Germain Prime search, the new discoveries are 388,342 digits long, smashing the previous twin prime record of 200,700 digits.

PrimeGrid is a collaborative project (similar to GIMPS, which searches for specifically Mersenne Primes) in which anyone who downloads their software can donate their unused CPU time to prime searching. It’s been the source of many recent prime number discoveries, including several in the last few months which rank in the top 160 largest known primes.

The University of Tennessee Martin’s Chris Caldwell maintains a database of the largest known primes, to which the new discovery has been added.

Further Reading

Press release from PrimeGrid (PDF)
The List of Largest Known Primes
PrimeGrid website
The new twin primes’ entries on the List of Largest Known Primes: n+1, n-1