Ramanujan Notes

Why are prime numbers interesting?

• Mathematicians are interested in the prime numbers because they form the building blocks for the natural numbers in the same way that atoms form the building blocks for more complex molecules. Every natural number can be factored into a product of prime numbers, the prime numbers being the indivisible "atoms" of our number system.
• They have fascinated mathematicians for centuries
• They are a corner stone of secure web sites, Electronic commerce and privacy protected E-Mail
• Ramanujan and Primes

Ramanujan was very interesting in prime numbers and patterns.

Ramanujan found a proof of Chebyshev's theorem that there is always a prime between a number and its double. For example, gien any number, say 2, we double it (4). We can always find a prime number between those two numbers (3 in this case). Others were taken with the beauty of Ramanujan's approach. Since Ramanujan did not have standard mathematical training, he did not know that this had already been proven by Chebyshev. Ramanujan's approach to the proof was very different than Chebyshev's.

The Prime Number Theorem gives an estimate of the number of primes up to an integer n. As Gauss discovered, the estimate gets better as n gets larger.
 Integers below n Actual number of primes Gauss' estimated number of primes Percentage of error 1,000 168 145 16.0% 1,000,000 78,498 72,382 8.4% 1,000,000,000 50,847,478 48,254,942 5.4%
Of course, Gauss was working without a computer and so he would not have known the actual number of primes below very large numbers. The idea would be to use the estimates in order to get a sense of the number of primes below a certain point.

Ramanujan thought he had improved on Gauss' estimates, but he had made a mistake. But he did come up with formulas for all sorts of other things related to primes that were correct and that Gauss didn't even consider.