The Mathematicians’ Million Dollars. It’s waiting for you…

Henri Poincare (1854-1912) was one of the most eminent French mathematicians of the past two centuries.

One of Poincare’s best-known problems is what is today called the Poincare conjecture.

The poincare conjecture is considered so important that the clay Mathematics Institute named it one of the seven millennium prize problems will be awarded $1 million.

The Poincare conjecture falls within the realm of topology.

This branch of mathematics focuses, roughly speaking, on the issue of whether one body can be deformed into a different body through pulling, squashing or rotating, without tearing or gluing pieces together.

A ball, an egg, and a flowerpot are, topologically speaking, equivalent bodies, since any of them can be deformed into any of others without performing any of the “illegal” actions.

A ball and a coffee cup, on the other hand, are not equivalent, since the cup has a handle, which could not have been formed out of the ball without poking a hole through it.
The ball, egg, and a flowerpot are said to be “simply connected” as opposed to the cup, a bagel, or a pretzel.

Poincare sought to investigate such issues not by geometric means but through algebra, thus becoming the originator of “algebraic topology.”

In 1904 he asked whether all bodies that do not have a handle are equivalent to spheres. In two dimensions this questions this question refers to the surfaces of eggs, coffee cups, and flowerpots and can be answered yes. (Surfaces like the leather skin of a football or the crust of a bagel are two-dimensional objects floating in three-dimensional space)

For three-dimensional surfaces in four-dimensional space the answer is not quite clear. While Poincare was inclined to believe that the answer was yes, he was not able to provide a proof.

Several Mathematicians were able to prove the equivalent of Poincare’s conjecture for all bodies of dimension greater than four. This is because higher-dimensional spaces provide more elbowroom so mathematicians find it simpler to prove the Poincare conjecture.

But, for three-dimensional surfaces in four-dimensional space –remember: The surface of a four-dimensional object is a three-dimensional object. Poincare’s conjecture remained as elusive as ever.

See you later
Carlos Tiger without Time

 

How the twin primes to cause the error in the processor Pentium Intel?

How the twin primes to cause the error in the processor Pentium Intel?
Whitin the group of integers, prime numbers are in a way thought of as atoms, since all integers can be expressed as a product of prime numbers (for example, 30=2x3x5), just as molecules are made up of separate atoms.

The theory of prime numbers continues to be shrouded in mystery and still holds many secrets.

Taking the first 100 numbers we count 25 primes; between 1001 and 1100 there are only 16; and between the numbers 100,001 and 100100 there are a mere six.

Prime numbers become increasingly sparse. In other words, the average distance between two consecutive primes becomes increasingly large.

Around the turn of the 19th century, the Frenchman Adrien-Marie Legendre and the German Carl Friedrich Gauss studied the distribution of primes. Based on their investigations they conjectured the space between a prime P and the next bigger prime would on average, be as big as the natural logarithm of P.

Sometimes the gaps are much larger, sometimes much smaller. There are even arbitrarily long intervals in which no primes occur whatsoever. The smallest gap. On the other hand, are two, since there is at least one even number between any two primes.

Primes that are separated from each other by a gap of only two –for instance, 11 and 13, or 197 and 199- are called twin primes.

There are also prime cousins, which are primes separated from each other by four nonprime numbers. Primes that are separated from each other by six nonprime numbers are called, sexy primes.

Much less is known about twin primes than about regular primes. What is certain is that they are fairly rare.

Among the first million integers there are only 8169 twin prime pairs. The largest twin primes so far discovered have over 50,000 digits. But much is unknown.

Nobody knows whether an infinite number of twin prime pairs exist, or whether after one particular twin prime pair there are no larger ones.

Working on the theory of twin primes, Thomas Nicely from Virginia, in the 1990s. He was running through all integers up to 4 quadrillion.

The algorithm required the computation of the banal expression X times (1/X). But to his shock, when inserting certain numbers into this formula, he received not the value 1 but an incorrect result.

On October 30, 1994, his computer consistently produced erroneous results when calculating the above equation with numbers ranging between 824,633,702,418 and 824,633,702,449. Through his research on twin primes, Thomas Nicely had hit on the notorious Pentium bug.

The error in the processor cost Intel, the manufacturer, and $500 million in compensations.

See you later
Carlos Tiger without Time

 

Languages spoken in the world and Mathematical poetry

There are about 1,500 different languages spoken in the world today.

In the early 1940’s when it was first being organized, officials (ONU) proposed that all diplomats be required to speak a single language, a restriction that would both facilitate negotiations and symbolize global harmony.

Over the years, there have been no fewer than 300 attempts to invent and promulgate a global language, the most famous being made in 1887 by the polish oculist L.L. Zamenhof. The artificial language he created is called Esperanto, and today more than 100,000 people in twenty-two countries speak it.

United Nations ambassadors are now allowed to speak any one of five languages: Mandarin Chinese, English, Russian, Spanish,  or French.

Today who speak mathematics fluently, as measured by the millions and by the historic consequences of their unified efforts, is arguably the most successful global language even spoken.

Though it has not enabled us to build a tower of Babel, it has made possible achievements that once seemed no less impossible: electricity, airplanes, the nuclear bomb, landing a man on the moon, and understanding the nature of life and death.

Matthe Arnold said: “ Poetry is simply the most beautiful, impressive, and widely effective mode of saying things.”

In the language of mathematics, equations are like poetry: They state truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the initiated to comprehend. And just as conventional poetry helps us to see deep within ourselves, mathematical poetry helps us to see far beyond ourselves – if not all the way up to heaven, then at leapt out to the brink of the visible universe.

In attempting to distinguish between prose and poetry, Robert Frost once suggested that a poem, by definition, is a pithy form of expression that can never be accurately translated. The same can be said about mathematics: It is impossible to understand the true meaning of an equation, or to appreciate its beauty, unless it is read in the delightfully quirky language in which it was penned.

· Summarized and adapted of “Mathematical Poetry” of Dr. Michael Guillen