A new discovery about prime numbers is that they are not truly random. Prime numbers that are not factorable by 2 or 5 only end in 1,3,7or 9. If prime numbers were truly random we would expect a prime number ending in 1 to be followed by another prime ending in 1 about 25% of the time but apparently using the first billion prime numbers the likelihood is more like 18%. This is a bias and they discovered biases for prime numbers ending in 3,7 and 9 as well.
Am I missing something here? Don't we know that there is a bias?
We know that any prime not factorable by 2, 3 or 5 = 30m + 1 or 7 or 11 or 13 or 17 or 19 or 23 or 29. Surely we can work out biases from this, for example a number ending in a number is never followed by another number ending in the same number. I.e. A number ending in 1 never follows another number ending in 1. This means the other numbers are more likely. And that's what Soundararajan and Oliver discovered; an 'anti-sameness' bias.
"Time not important. Only life important." - The Fifth Element 1997
How do you define random?
For a lucid summary of the topic:
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Well, I guess it depends on what you mean by 'random'. The primes are not really randomly distributed, though they may appear that way. Of course, there is Chebyshev's bias which you are alluding to, and the Prime Number Theorem, one of the most celebrated theorems in mathematics, which gives you an asymptotic expression for the number of primes less than a given magnitude, and also a similar asymptotic formula for the nth prime.
This is a bias and they discovered biases for prime numbers ending in 3,7 and 9 as well.
Chebyshev's bias tends to correct itself asymptotically.
Last edited by zetafunc (2016-10-30 21:54:37)