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**Primenumbers****Member**- Registered: 2013-01-22
- Posts: 141

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.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

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.**

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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 *n*th 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)*

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**KenE****Member**- Registered: 2017-06-17
- Posts: 5

The problem with trying to find a pattern for the prime numbers is that everyone leaves the even numbers on the "sieve". I finally hit on the idea to leave out the even numbers and only list the odd numbers on my "sieve". I had to leave the 5 composites in the list since there are four numbers between 3 and 7 and two between 1 and 3 and two between 7 and 9. The results were very interesting. Got to go to work. I will try to explain later.

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**KenE****Member**- Registered: 2017-06-17
- Posts: 5

I finally decided to use a graph that had the odd numbers from 1 to 49 in the top row. I the second row, 51 to 99. I continued down to the last row ending in 349. After marking down all the composites of 3, down to 51, I went to 3 on the top row. Dropping down to the next row (essentially making it 53 and subtracting 2, I arrived at 51, 3*17. at 15 +50 -2, I arrived at 63. Every composite of 3 was add 48, or add 54 to go the other way. The 7 composites were add 42, or add 56 for the direction you want to go. This ran true for my graph up to 349. If the top row is changed to 1 to 99, 1 to 149,or higher, the pattern for each prime composite changes. Ex. 3 + 100= 103 +2 =105. Useless information, but interesting.

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**Jabuto****Member**- Registered: 2017-07-29
- Posts: 11

That explanation imo is little off.

You should show that it is biased in every number system N. Decimal system is largely arbitrary.

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**KenE****Member**- Registered: 2017-06-17
- Posts: 5

The "revised sieve" seems to work with any top row that is devised of multiples of 10 (1-9 all odd numbers included, 1-99 all odd numbers included) , and on up. Yes, changing the numbers on the top row will change the pattern for each composite formed by a specific prime. All of the composites formed by any specific prime start at 3*P(or 3*C) if you want to and work up the number ladder(3*1,3*3,3*5,and up through every odd number in sequence for all composites formed by 3). if you start with 5*1 your first composite is 5*3, and on up the line in sequence. Choose any prime you want (you can choose any composite also) and follow the list. It will start at 1*P or 1*C and move to 3*P or C, 5*P or C, 7* P or C, 9*P or C, and on up in sequence(11,13,...the last number). I ask you, or anyone else, to try my method to prove if I am right or wrong before dismissing it out-of-hand that I am wrong. Is there anyone willing to take the challenge? The composites formed by 3 are easy to find, as they will move forward or back 1 odd number, depending on the length of the top row. The composites formed 5 run straight down the list if you use a top row of multiples of 10 (5,15,25,...). The 7 composites will run, on the row below, forward or back 1-6 spaces (depending on the length of the top row) for a total of 7 odd numbers(forward move+ back move=7 odd numbers.

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,332

random is an excuse for unable to find the true cause.

**X'(y-Xβ)=0**

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