|
|
|
Choosing, Cleansing and Charging Crystals
I have written a lot of articles about gemstones, rocks and crystals for this site that describe which ones are best for prosperity, love and protection. If you look back in the archives here, you will find lots of information about gemstones,...
Depression
The 'experts' have tried to evaluate me as a sufferer of this but found no such evidence or behavior. The same was true for all other mental or medical conditions, so they think 'witch' covers my condition in life; because I refuse to participate in...
Elevator Action
I thank Science for the gift of a reliable friend who lifts me
up and escorts me down with just one button push. Elevators take
people, as well as their baggages to wherever it is programmed
to go.
Elevators are enclosure or platforms that...
It's a Quantum Thing
We don't need to understand quantum physics entirely in order to appreciate it. Even those who have devoted their lives to the study of the universe and its atomic structure will admit that many mysteries remain. Well, I love mysteries, so let's...
Quotes to Think About
INSPIRATIONAL COMMENTS: “There is a principle which is a bar against all information, which is proof against all arguments and which cannot fail to keep a man in everlasting ignorance – that principle is contempt prior to investigation. – Herbert...
|
|
| |
|
|
|
|
|
|
The “beauty” of number theories
In the proceedings of the London Mathematical Society of May 1921, Srinivasa Ramanujan said something startling as a reply to G.H. Hardy’s suggestion that the number of a taxi-cab (1729) was “dull.” Ramanujan’s reply was as follows:
“No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”
“Wow,” I remember thinking when I first read that. And he was right of course. Ramanujan spotted the two sets of cubes being 13 +123 and 93+103 respectively. Grab the nearest calculator and check the results of these two expressions. Scary?
Ramanujan is today considered as the father of modern number theory by many mathematicians. He made a number of original discoveries in number theory, especially in collaboration with G.H Hardy, a theorem concerning the partition of numbers into a sum of smaller integers. For example the number 4, has five partitions as it can be expressed in five ways which are ‘4,’ ‘3+1,’ ‘2+2,’ ‘2+1+1’ and ‘1+1+1+1.’
Ramanujan made partition lists for the first 200 integers in his tattered notebook and observed a strange regularity. For any number that ends with the digit 4 and 9, the number of possible partitions is always divisible by 5. Secondly, starting with the number 5, the number of partitions for every seventh integer is a multiple of 7. And thirdly, starting with 6, the partitions for every eleventh integer are a multiple of 11.
These strange numerical relationships that Ramanujan discovered are now called the three Ramanujan “congruences.” And these relationships completely shocked the mathematical community: the multiplicative behaviours should apparently have had nothing to do with the additive structures involved in partitions.
However during the Second World War, Freeman Dyson, a mathematician and physicist, developed a tool that allowed him to break partitions of whole numbers into numerical groups of equal sizes. According to Dyson, this tool, which he called “rank,” would be able to prove the three Ramanujan congruences. Unluckily for Dyson though, rank worked only with 5 and 7 but not with 11. He had however made a huge step in proving Ramanujan’s findings. But the problem now seemed to be
with 11, right? Partly!
In the 1980s, the mystery of 11 was finally solved by 2 other mathematicians, Andrews and Garvan. But the story did not end there.
In the late 1990s, completely by chance, Ken Ono, and expert on Ramanujan’s work, came upon one of Ramanujan’s original tattered notebooks. In there he notice a peculiar numerical formula that seemed to have no link whatsoever with partitions. The formula however proved to be the spinner.
Working with the formula, Ono proved later that partition congruences do not only exist for 5,7 and 11 but could also be found for all larger primes. So now would Andrews’s and Garvan’s tool (called “crank”), inspired from Dyson’s “rank,” work with all those infinite number of partition congruences?
Well yes. Karl Mahlburg, a young mathematician has spent a whole year manipulating numerical formulae and functions that came out when he applied “crank” on various prime numbers. Mahlburg says he slowly started spotting uniformity between the formulae and functions. And then came the brainstorm or “fantastically clever argument,” as Ono puts it.
Basing his owns work on Ono’s, Mahlburg discovered that the partition congruence theorem still holds if the partitions were broken down in a different manner. Instead of breaking the number 115 for example into five equal partitions of 23 (which is not a multiple of 5), he split the number into 25, 25, 25, 30 and 10. As each part is a multiple of 5, it follows that the sum of the parts is also a multiple of 5. In fact Mahlburg showed that this concept extends to every prime number thereby proving that Andrews’s and Garvan’s crank worked for all those infinite number of partition congruences.
Incredible how such number theories, which are so full of wit, may be as difficult as this to prove. It did take about a complete century to prove Ramanujan’s partition congruences anyway.
But coming back to the taxi-cab number 1729. This number is nowhere near “dull.”
(9-7)/2=1
9-(2/1)=7
(9/1)-7=2
(7/1)+2=9
Cool!
About the Author
Khalil A. Cassimally is currently Senior Columnist at BackWash.com and Columnist for bbc.co.uk h2g2 The Post where he writes 'Not Scientific Science' column.
|
|
|
|
|
| Science/AAAS | Scientific research, news and career information |
| International weekly science journal, published by the American Association for the Advancement of Science (AAAS). |
| www.sciencemag.org |
|
| Science/AAAS | Table of Contents: 1 December 2006; 314 (5804) |
| This Week in Science: Editor summaries of this week's papers. Science 1 December 2006: 1349. ... 2006 American Association for the Advancement of Science. ... |
| www.sciencemag.org |
|
| Science.gov : FirstGov for Science - Government Science Portal |
| Science.gov is a gateway to government science information provided by US Government science agencies, including research and development results. |
| www.science.gov |
|
| ScienceDaily: Your source for the latest research news and science ... |
| ScienceDaily -- the Internet's premier online science magazine and science news web site -- brings you the latest discoveries in science, health & medicine, ... |
| www.sciencedaily.com |
|
| Science News - New York Times |
| Find breaking news, science news & multimedia on biology, space, the environment, health, NASA, weather, drugs, heart disease, cancer, AIDS, mental health ... |
| www.nytimes.com |
|
| Science News Online |
| Weekly magazine offers featured articles from the current issue along with special online-only features. Includes photo collection, archives, ... |
| www.sciencenews.org |
|
| Science in the Yahoo! Directory |
| Explore the fields of astronomy, biology, geology, mathematics, and physics and all of their related disciplines with resources designed for professionals, ... |
| dir.yahoo.com |
|
| Open Directory - Science |
| Agriculture (2454); Anomalies and Alternative Science (525); Astronomy (4208); Biology (20593); Chemistry (4852); Computer Science@ (2358) ... |
| dmoz.org |
|
| BBC - Science & Nature |
| The best of BBC Science and Nature, from TV and radio, to the web and beyond. Take a tour from the smallest atoms, to the largest whales and the most ... |
| www.bbc.co.uk |
|
| Science - Wikipedia, the free encyclopedia |
| Sciences versus Science: the plural of the term is often used but is difficult to ... Science education is also a very vibrant field of study and research. ... |
| en.wikipedia.org |
|
| Popular Science |
| Monthly magazine about current science and technology. |
| www.popsci.com |
|
| Science/AAAS | ScienceNOW: The Latest News Headlines from the ... |
| AAAS web magazine. Some free sample stories, subscription required for full text. |
| sciencenow.sciencemag.org |
|
| ScienceCareers.org | Science Jobs, Funding, Meetings, and Advice ... |
| Searchable database of jobs, sorted by field specialty. Can post resume and curriculum vitae. Includes tips for improving the workplace for employers and ... |
| sciencecareers.sciencemag.org |
|
| American Association for the Advancement of Science |
| Research news, issue papers. Educational programs, science policy (US and international). |
| www.aaas.org |
|
| NASA - Science@NASA |
| News and features about NASA research, aimed at the general public. Includes sections on astronomy, space science, beyond rocketry, living in space, ... |
| science.nasa.gov |
|
| Science NetLinks: Resources for Teaching Science |
| Resources for K-12 science educators. |
| www.sciencenetlinks.com |
|
| Cool Science for Curious Kids |
| Fun and interactive site to help kids appreciate science. Why are snakes like lizards, and monkeys like moose? Find out here. |
| www.hhmi.org |
|
| Welcome to the Science Museum |
| London museum and library of science. Exhibitions cover all areas of science and technology. Includes online exhibits and a learning area. |
| www.sciencemuseum.org.uk |
|
| New Scientist - International News, Ideas, Innovation |
| Weekly science and technology news magazine, considered by some to be the world's best, with diverse subject matter. Articles from current issue and ... |
| www.newscientist.com |
|
| CNN.com - Science and Space |
| Offers news stories related environmental issues, archeology, astronomy, technology, geology and other science topics. |
| www.cnn.com |
|
|