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Thursday, August 28, 2014

Throwback Fact of the Week - Sieve of Eratosthenes - 8/28/14

Sieve of Eratosthenes

Quickly recall that a prime number is a number larger than 1 that is divisible only by itself and 1 (e.g. 2, 5, 7, 11...). Euclid showed that there are an infinite number of primes. However, it wasn't until around 240 B.C. that Eratosthenes developed the first known method for finding primes, a method known today as the Sieve of Eratosthenes.

The Sieve can be used to find all prime numbers up to a given integer, n

The method is straightforward: make a a list of all the integers less than or equal to n, cross out the multiples of all primes that are less than or equal to the square root of n, the numbers that remain are the primes. Typically it is easiest to start with 2, cross out all multiples of 2. Then 3, cross out all its multiples etc.

An example with n = 120 is demonstrated the video below. The video crosses out the multiples of 2, then of 3, then of 5 and lastly the multiples of 7 (since 7 is the largest prime that is less than the square root of 120). After 7, the video highlights the remaining primes in pink.

 By Ricordisamoa (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Prime numbers continue to fascinate mathematicians today. Some of the most famous unsolved problems in mathematics center around prime numbers, including the Goldbach Conjecture and the Riemann Hypothesis.

Thursday, August 21, 2014

Throwback Fact of the Week - Perfect Numbers - 8/21/14

Perfect Numbers

perfect number is a positive integer that is equal to the sum of its proper positive divisors. In other words, it is equal to the sum of its positive divisors excluding the number itself. The simplest example of this is the number 6; the proper divisors of 6 are 1, 2, and 3, 1+2+3=6, so 6 is a perfect number.

Knowing that 6, which is a small integer, is a perfect number might lead one to believe that numbers of this form are common. This turns out to be false; they are much rarer. The next perfect numbers are 28, 496, and 8128. The next perfect number after 8128? 33,550,336.

Perfect numbers are not new. Pythagoras and the Greeks knew about the first four perfect numbers (you can see how they might not have encountered or been able to find the factors of a number like 33,550,336).

There have been a number of interesting results about perfect numbers. The most important result was initially discovered by Euclid and later expanded upon by Euler.

Euclid proved that if if 2p-1 is prime, then N = (2p-1)(2p-1) is an even perfect number.

Prime numbers of the form 2p-1, where p is prime, are known as Mersenne primes, another bountiful world of study themselves. Two millennia later, Euler proved that (2p-1)(2p-1), where 2p-1 is a Mersenne prime, will produce all even perfect numbers (there is a 1-1 ratio).
It remains unknown if there are an infinite number of Mersenne primes (only 43 Mersenne primes are known so far). It also remains unknown if there are any odd perfect numbers.

Tuesday, August 19, 2014

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Thursday, August 14, 2014

Throwback Fact of the Week - This Day in History: Guido Castelnuovo - 8/14/14

On this day in history

On August 14th, 1865 Guido Castelnuovo was born in Venice, Italy. He was an Italian mathematician who made contributions to the fields of algebraic geometry and statistics.

Despite being forced into hiding during World War II, Castelnuovo organized and taught courses to Jewish students who were forbidden from attending university.

After the war, he was charged with repairing the damage done to Italian institutions throughout the twenty years of Mussolini's rule.

Thursday, August 7, 2014

Throwback Fact of the Week - Golden Ratio - 8/07/14

The golden ratio is a deceptively intriguing bit of mathematics. The ratio makes numerous appearances throughout history in architecture, nature, art, music, and most importantly mathematics.

It is easiest to understand the ratio by imagining a line. Divide the line into two segments with long segment a and shorter segment b. The segments are said to be in the golden ratio if the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part, or simply if (a+b)/a = a/b. This ratio comes out to be an irrational number (approx. 1.61803...), known as the golden ratio, and is symbolized by the Greek letter φ (phi).

If the lengths of the sides of a rectangle are in the golden ratio, the rectangle is called a golden rectangle. This rectangle exhibits some interesting properties. It is possible to divide a golden rectangle into a square and another (smaller) golden rectangle (pictured below). This process can be repeated indefinitely.

A logarithmic spiral can be drawn and this spiral will closely approximate the Golden Spiral, which is a spiral that gets wider by a factor of φ with ever quarter turn.