### Sunday, March 14, 2010

## JSH: Twin primes probability correlation

With twin primes a simple approach rips the prime distribution out of the equation.

My twin primes probability probability calculation works by taking the ACTUAL COUNT of prime numbers in the interval p_{j-1}^2 to p_j^2, and multiplying times:

prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*…*(1/2)

The result is easy as it is just multiplying the probability for each prime x in that interval that it is NOT true that

x + 2 ≡ 0 mod p

which probability is just the result of dividing one minus the number of non-zero residues by the total number of residues together to get the total probability that a prime plus 2 is also prime.

For example, between 5^2 and 7^2, there are 6 primes. The probability then is given by:

prob = ((5-2)/(5-1))*((3-2)/(3-1) =3D (3/4)*(1/2) =3D 0.375

And 6*0.375 =3D 2.25 so you expect 2 sets of twin primes in that interval. And those twin primes are 29, 31 and 41, 43.

(Compare the simplicity of the calculation with traditional approaches which leave in the prime distribution itself which generates a LOT more complexity.

See: http://mathworld.wolfram.com/TwinPrimes.html)

Since the result equals prediction correlation is 1.

But that's just one example, and you need to do a lot of them.

Within the bigger picture I've also predicted a smaller impact related to the distance p_j - p_{j-1}. Where I have hypothesized that the greater that distance the greater the accuracy of the prediction, which is also a mathematically precise statement as "accuracy" has a mathematical meaning in this context as does "correlation".

My twin primes probability probability calculation works by taking the ACTUAL COUNT of prime numbers in the interval p_{j-1}^2 to p_j^2, and multiplying times:

prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*…*(1/2)

The result is easy as it is just multiplying the probability for each prime x in that interval that it is NOT true that

x + 2 ≡ 0 mod p

which probability is just the result of dividing one minus the number of non-zero residues by the total number of residues together to get the total probability that a prime plus 2 is also prime.

For example, between 5^2 and 7^2, there are 6 primes. The probability then is given by:

prob = ((5-2)/(5-1))*((3-2)/(3-1) =3D (3/4)*(1/2) =3D 0.375

And 6*0.375 =3D 2.25 so you expect 2 sets of twin primes in that interval. And those twin primes are 29, 31 and 41, 43.

(Compare the simplicity of the calculation with traditional approaches which leave in the prime distribution itself which generates a LOT more complexity.

See: http://mathworld.wolfram.com/TwinPrimes.html)

Since the result equals prediction correlation is 1.

But that's just one example, and you need to do a lot of them.

Within the bigger picture I've also predicted a smaller impact related to the distance p_j - p_{j-1}. Where I have hypothesized that the greater that distance the greater the accuracy of the prediction, which is also a mathematically precise statement as "accuracy" has a mathematical meaning in this context as does "correlation".