Published January 28, 2016
Tags: Algorithms, Number Theory
Once again the record for the largest prime number has been shattered. As with all recent records, the new number is a Mersenne prime, a number of the form
Mp = 2p – 1
where p itself is a prime. Participants in a distributed computing project called GIMPS (Great Internet Mersenne Prime Search) continue without rest to search for ever-larger primes of this form.
Most of the recent large primes have been found in the GIMPS project (for an earlier post on GIMPS, click Mersennery Quest. The project uses a search algorithm called the Lucas-Lehmer primality test, which is particularly suitable for finding Mersenne primes. The test, which was originally devised by Edouard Lucas in the nineenth century and extended by Derek Lehmer in 1930, is very efficient on binary computers.
Continue reading ‘Prime Number Record Smashed Again’
Published January 21, 2016
Tags: Algorithms, Music
An ingenious method of tuning pianos, based on the concept of entropy, has recently been devised by Haye Hinrichsen of Würzburg University. Entropy, which first appeared in the mid-nineteenth century in thermodynamics and later in statistical mechanics, is a measure of disorder. Around 1948 Claude Shannon developed a mathematical theory of communications and used entropy as an indicator of information content [TM084, or search for “thatsmaths” at irishtimes.com].
Tuning curve showing the stretch for high and low notes (Image: Wikimedia Commons).
Continue reading ‘Entropy Piano Tuning’
Published January 14, 2016
Tags: Analysis, Probability
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.
Continue reading ‘Twin Peaks Entropy’
Modern weather forecasts are made by calculating solutions of the mathematical equations that express the fundamental physical principles governing the atmosphere [TM083, or search for “thatsmaths” at irishtimes.com]
The solutions are generated by complex simulation models with millions of lines of code, implemented on powerful computer equipment. The meteorologist uses the computer predictions to produce localised forecasts and guidance for specialised applications.
Continue reading ‘Richardson’s Fantastic Forecast Factory’