Prime Time

sieve:  a utensil with meshes or holes to separate finer particles from coarser ones or solids from liquids. – Merriam Webster Dictionary

Every student who has completed grade school learned about prime numbers.  We learned what they are:  natural numbers greater than 1 with factors of only 1 and themselves.  We learned methods to find them.  We learned to use them in factoring.  An elementary exposure was part of the ticket out of elementary mathematics.

Not everyone, though, was exposed to the Sieve of Eratosthenes.  The Sieve is an ancient method (circa 240 B.C.) that gives us a systematic way to find primes, starting with 2.  You write the natural numbers from 1 up to however high you want to go.  You cross 1 out, as it is neither prime nor composite.  Start with 2, your first prime.  (I usually have students circle it.)  Then, cross out all multiples of 2, as they have additional factors other than 1 and themselves.  Go back, and the next available number, 3, is prime.  Circle it.  Cross out all multiples of 3.  Go back, and the next available number, 5, is prime.  Circle it.  Cross out all multiples of 5.  Go back.  Repeat over and over until you reach the end of your sieve.

The method is pretty simple in its execution, but it’s time-consuming when trying to sort out primes for large numbers.  Even with today’s super computers, the method takes up too much memory and time.  But, there’s hope!  Peruvian mathematician Harald Helfgott has found a way to modify the sieve so that computers don’t need to use as much memory space.  Scientific American published an article on the discovery, and it is worth a read.

New Take on an Ancient Method Improves Way to Find Prime Numbers

Nearly Twin Primes

Twin primes are primes that occur is succession with one composite between them.  For example, 3 and 5 are twin primes.  They are primes in that they only have factors 1 and themselves, and then they are twins because they are separated only by one composite which is 4.  The Twin Prime Conjecture claims that we can find this sort of pattern to infinity.  No matter how far we go out on the number line, there will be another pair of twin primes to find.  To view a really cheesy but fun video about this, check out the NOVA Science Now link.

Recently, the math world has exploded with discovery concerning this conjecture.  Mathematicians had not been able to prove the Twin Prime Conjecture exactly as is, but they were able to prove that primes go on in a skippy pattern with more than one composite between them.  Efforts are being made to narrow the gap, and hopefully eventually prove the gap of only one.

In May of 2013, it was revealed that Tom Zhang of the University of New Hampshire proved infinite pairs of primes that are 70 million apart.  This seems like a big gap, but it was remarkable as it was the closest anyone had come.  (It is definitely a smaller gap than the previous infinity.)  To make it even more astounding, Professor Zhang was relatively unknown until this.  He has been quietly working on his mathematics for years as he went about life and work in jobs such as sandwich maker for Subway.

The big reveal of Dr. Zhang’s discovery started a flurry of work.  Dr. Terry Tao started a Polymath Project to enlist many volunteers to continue work on this project.  The team narrowed down the gap from 70 million to 4,680 by July.

As of September, James Maynard had reduced the gap to 600.

In November, Thomas Engelsma says he reduced the gap to 576.

And the work goes on!