New Information On Prime Numbers.

 My apologies.  I was traveling with one computer and it died while out of the country so I didn't get anything done for Monday.  Now on to today's topic.  I remember having a math professor who did two things.  One was when he wrote his tests, he did it in a bar and wrote the problems on a napkin.  The second thing he continually worked on was finding the largest prime number he could.  I think he retired before he did that.

Now there is a new generation of mathematicians who are working on figuring out the distribution of prime numbers and understanding that distribution. Remember that Eratosthenes came up with the first real method of finding prime numbers between one and 100 known as Eratosthenes's sieve.  There are sieves to find all sorts of primes such as twin primes, etc.

Although we have the sieves, people do not understand the distribution of primes.  Since it is much harder to find primes in a sieve that contains much larger numbers, mathematicians have to estimate the number of primes they think are in the range. In addition, it is often harder to make predictions when looking at other sieves such as the ones for twin primes due to the size of the remainders.  

As you move to larger numbers, it has been found that the remainders fall into a statistically predictable pattern and eventually even out. This means say you take the remainders of 1 and 2 when divided by 3 and place them in one of two buckets. eventually, the two buckets will have the same number of primes.  Mathematicians need to know when the buckets even out and how soon that happens in order to know more about primes.

There were spurts in investigation in the 60's and 80's but nothing more happened until recently. A mathematician investigated the question about buckets evening out and how soon that happens, and calculated that the level of distribution was 0.6 for commonly used sieves. 

His grad students extended it to 0.617.  To do this, they used a technique of inclusion/exclusion which is similar to what students do when they work with the sieve of Eratosthenes.  They exclude 2 and all it's multiples which eliminates about half the numbers or 50%.  Then they exclude 3 and all its multiples which throws out another 1/3rd of the numbers. Ok, this means that due to the way things are counted, many numbers are double counted such as 6 and its multiples because 2 and 3 are factors of 6.  So you add the 1/2 + 1/3 and then subtract 1/6 to account for the twice counted numbers.  This way you do not have an over estimation.

So then you eliminate all numbers for 5 and its multiples but you have to subtract 1/10 and 1/15 to account for any numbers double counted due to 2 and 3.  Thus the process continues with the denominators getting bigger and bigger.  This creates and upper and lower boundary rather than an exact answer. 

This lead to someone to propose the idea that the buckets even out based on the generalized Riemann Hypothesis. This means we are looking for all the primes up to N and the remainders are equally divided up into the number of buckets equal to the square root of N.  So this opens the way for more possibilities in determining the number of primes but it will be a while.  

Let me know what you think, I'd love to hear.  Have a great day.