Showing posts from October, 2012

Partitions and Dynamic Programming

There are exactly 7 ways to write the number 5 as a sum of other positive numbers: 5 = 5 1+4 = 5 1+1+3 = 5 1+1+1+2 = 5 1+1+1+1+1 = 5 2+3 = 5 2+2+1 = 5 There is a convention to say that P(5) = 7, meaning that there are 7 partitions of the number 5. Such a notion goes back to the times of Euler, in the 1700’s. At that time, in 1735, an amateur mathematician managed to find the first N for which P(N) had over 100 digits. As far as I know there were no laptops around during those days, hence it is truly marvelous what that young man accomplished. Unfortunately his name and story was lost in time, and we only know about this tale thanks to one side note from Euler himself in one of his books. Of course, today with the advanced of powerful computers, it is possible to find way bigger numbers. For instance, using an average laptop, it is possible to find the first number N for which P(N) has 200 or more digits:   P(N) = 10058155246188730579133793592616739522467948239017058084

The Nazca Primes

The Nazca Lines are considered one of the most intriguing mysteries of ancient societies. They consist of several drawings of gigantic images, mostly animals and images resembling humans, created by the Nazca culture, around 650 AD. The images are so big that they only make sense when observed from extremely high altitudes: “The Astronaut”, in the Nazca Desert (source: Bing Images) The purpose of the hundreds of such drawings remains a mystery. Scientists and anthropologists continue their studies and excavations in order to solve this mystery. But in August 2007, a group of Israeli and Iranian anthropologists made a very surreal discovery when excavating the caves of the Nazca Desert in southern Peru. Excavations of the caves in the Nazca Desert, August 2007 (source: Bing Images) Amongst several new (albeit small) drawings found in some of the caves in that region, one particular drawing caught the eye of the resea