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Are there More or Less stars than grains of beach sand?

This week’s episode of More or Less on the BBC World Service answered a question that involved estimating big numbers: Are there more stars than grains of beach sand?

This claim was famously made by Carl Sagan in the seminal programme Cosmos.

The cosmos is rich beyond measure. The number of stars in the universe is larger than all the grains of sand on all the beaches of the planet Earth.

More or Less come to a fairly standard answer, that Sagan was correct. This sort of problem, which involves approximating unknowable numbers based on a series of estimates, is called a Fermi problem. I’ve written about Fermi problems here before. The More or Less approach to answering this raised a question from a reader of this blog.

Alright, actually Paul is one of the writers of this blog, rather than a reader. Even so, are his concerns warranted?

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.

Happy Birthday to me

“Life moves very fast. It rushes from Heaven to Hell in a matter of seconds.”
― Paulo Coelho

This week, I was suddenly reminded of a fact I’d been meaning to keep track of, and I was disappointed to discover that even though I always endeavour to remember birthdays and holidays (mainly due to a system of elaborate reminders, notes and excessive list-making), I’d missed a hugely significant anniversary. Shortly after the clock struck midnight on New Year’s eve, I had passed one billion seconds old.