Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number that can be written in the form M_n = 2ⁿ − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes (sequence A000668 in OEIS) are 3, 7, 31, and 127.

If n is a composite number then so is 2ⁿ − 1. (2^ab − 1 is divisible by both 2^a − 1 and 2^b − 1.) The definition is therefore unchanged when written M_p = 2^p − 1 where p is assumed prime.

More generally, numbers of the form M_n = 2ⁿ − 1 without the primality requirement are called Mersenne numbers. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those pernicious numbers whose binary representation contains no zeros. The smallest composite pernicious Mersenne number is 2¹¹ − 1 = 2047 = 23 × 89.

Mersenne primes M_p are also noteworthy due to their connection to perfect numbers.

Brady Haran

Brady John Haran (born 18 June 1976) is an Australian independent film-maker and video journalist who is known for his educational videos and documentary films produced for BBC News and for his YouTube channels, such as Numberphile and Periodic Videos.

Career

Brady Haran studied journalism for a year before being hired by The Adelaide Advertiser. In 2002 he moved from Australia to Nottingham, United Kingdom. In Nottingham he worked for the BBC, began to work with film, and reported for East Midlands Today, BBC News Online and BBC radio stations.

In 2007, Haran worked as a filmmaker-in-residence for Nottingham Science City, as part of an agreement between the BBC and The University of Nottingham. His "Test Tube" project started with the idea of producing a documentary about scientists and their research, but he decided to upload his raw footage to YouTube; from that point "Periodic Videos" and "Sixty Symbols" were developed. Haran then left the BBC to work full-time making YouTube videos.

Surname

A surname or family name is a name added to a given name. In many cases, a surname is a family name and many dictionaries define "surname" as a synonym of "family name". In the western hemisphere, it is commonly synonymous with last name because it is usually placed at the end of a person's given name.

In most Spanish-speaking and Portuguese-speaking countries, two or more last names (or surnames) may be used. In China, Hungary, Japan, Korea, Madagascar, Taiwan, Vietnam, and parts of India, the family name is placed before a person's given name.

The style of having both a family name (surname) and a given name (forename) is far from universal. In many countries, it is common for ordinary people to have only one name or mononym.

The concept of a "surname" is a relatively recent historical development, evolving from a medieval naming practice called a "byname". Based on an individual's occupation or area of residence, a byname would be used in situations where more than one person had the same name.

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ₁(n) = 2n.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby is an even perfect number whenever is what is now called a Mersenne prime—a prime of the form for prime Much later, Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

Examples

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in OEIS).

Mathematical Sciences Research Institute

The Mathematical Sciences Research Institute (MSRI), founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions. The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the hills overlooking Berkeley.

History

MSRI was founded in 1982 by Shiing-Shen Chern, Calvin Moore, and Isadore M. Singer. MSRI hosts about 85 mathematicians and postdoctoral research fellows each semester for extended stays and holds programs and workshops, which draw approximately 2,000 visits by mathematical scientists throughout the year. Unlike many mathematical institutes, it has no permanent faculty or members, and its scientific activities are overseen by its Directorate and its Scientific Advisory Committee, a panel of distinguished mathematicians drawn from a variety of different areas of mathematical research.