Class Records
Congruence Classes
The repartition of the first class records in the congruence classes for primes under `50` is shown in Table bellow
| Nb | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 |
| 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
| 0 | 174 | 379 | 196 | 151 | 93 | 75 | 57 | 42 | 47 | 27 | 37 | 24 | 32 | 23 | 20 | 15 |
| 1 | 826 | 280 | 197 | 128 | 95 | 83 | 56 | 37 | 51 | 30 | 28 | 21 | 35 | 25 | 19 | 24 |
| 2 | 341 | 215 | 150 | 106 | 86 | 61 | 37 | 42 | 33 | 36 | 20 | 19 | 27 | 19 | 19 | |
| 3 | 187 | 149 | 90 | 86 | 61 | 56 | 42 | 28 | 33 | 22 | 18 | 22 | 22 | 17 | ||
| 4 | 205 | 144 | 97 | 77 | 56 | 55 | 41 | 31 | 33 | 34 | 20 | 25 | 18 | 18 | ||
| 5 | 131 | 79 | 89 | 74 | 48 | 56 | 24 | 37 | 32 | 23 | 20 | 22 | 20 | |||
| 6 | 147 | 88 | 88 | 59 | 58 | 47 | 43 | 31 | 29 | 29 | 30 | 19 | 20 | |||
| 7 | 71 | 60 | 55 | 59 | 42 | 39 | 30 | 33 | 28 | 22 | 25 | 19 | ||||
| 8 | 101 | 87 | 66 | 58 | 42 | 22 | 31 | 34 | 30 | 32 | 13 | 22 | ||||
| 9 | 87 | 68 | 64 | 59 | 46 | 33 | 31 | 24 | 24 | 26 | 22 | 20 | ||||
| 10 | 93 | 71 | 58 | 49 | 51 | 42 | 29 | 17 | 25 | 25 | 21 | 18 | ||||
| 11 | 64 | 60 | 52 | 44 | 42 | 38 | 32 | 26 | 27 | 22 | 24 | |||||
| 12 | 66 | 60 | 61 | 32 | 44 | 30 | 28 | 23 | 26 | 22 | 24 | |||||
| 13 | 63 | 47 | 44 | 34 | 24 | 21 | 20 | 20 | 25 | 15 | ||||||
| 14 | 60 | 64 | 40 | 35 | 38 | 19 | 36 | 22 | 25 | 18 | ||||||
| 15 | 46 | 53 | 36 | 25 | 35 | 31 | 25 | 23 | 24 | 13 | ||||||
| 16 | 44 | 53 | 34 | 25 | 35 | 21 | 24 | 20 | 17 | 24 | ||||||
| 17 | 53 | 36 | 47 | 28 | 30 | 19 | 22 | 23 | 19 | |||||||
| 18 | 59 | 33 | 35 | 45 | 29 | 18 | 24 | 25 | 14 | |||||||
| 19 | 54 | 42 | 31 | 27 | 20 | 20 | 19 | 20 | ||||||||
| 20 | 49 | 28 | 28 | 23 | 16 | 18 | 25 | 18 | ||||||||
| 21 | 43 | 34 | 28 | 25 | 32 | 33 | 23 | 23 | ||||||||
| 22 | 48 | 34 | 42 | 23 | 23 | 17 | 19 | 20 | ||||||||
| 23 | 37 | 43 | 30 | 25 | 21 | 24 | 18 | |||||||||
| 24 | 48 | 25 | 32 | 19 | 21 | 20 | 18 | |||||||||
| 25 | 32 | 34 | 31 | 30 | 25 | 25 | 21 | |||||||||
| 26 | 31 | 23 | 27 | 28 | 26 | 21 | 20 | |||||||||
| 27 | 41 | 26 | 26 | 22 | 25 | 22 | 12 | |||||||||
| 28 | 34 | 22 | 21 | 30 | 26 | 16 | 18 | |||||||||
| 29 | 33 | 36 | 17 | 15 | 24 | 20 | ||||||||||
| 30 | 36 | 30 | 31 | 27 | 21 | 18 | ||||||||||
| 31 | 32 | 22 | 20 | 22 | 17 | |||||||||||
| 32 | 25 | 31 | 21 | 19 | 22 | |||||||||||
| 33 | 31 | 25 | 18 | 24 | 18 | |||||||||||
| 34 | 27 | 26 | 23 | 21 | 16 | |||||||||||
| 35 | 29 | 21 | 30 | 23 | 20 | |||||||||||
| 36 | 24 | 24 | 20 | 18 | 18 | |||||||||||
| 37 | 15 | 22 | 11 | 16 | ||||||||||||
| 38 | 19 | 21 | 20 | 26 | ||||||||||||
| 39 | 20 | 16 | 21 | 21 | ||||||||||||
| 40 | 30 | 29 | 21 | 14 | ||||||||||||
| 41 | 29 | 24 | 14 | |||||||||||||
| 42 | 16 | 14 | 18 | |||||||||||||
| 43 | 25 | 21 | ||||||||||||||
| 44 | 27 | 18 | ||||||||||||||
| 45 | 24 | 17 | ||||||||||||||
| 46 | 24 | 20 | ||||||||||||||
| 47 | 22 | |||||||||||||||
| 48 | 19 | |||||||||||||||
| 49 | 16 | |||||||||||||||
| 50 | 23 | |||||||||||||||
| 51 | 19 | |||||||||||||||
| 52 | 16 |
For each base except `2` and `3`, the hypothesis that the class records are equally distributed in each congruence class can be accepted with a significance level of 95%
Repartition in Basis 2
We find, amongst the 1000 first class records, 174 even values and 826 odd values.
We can evaluate the theoretical repartition between odd and even values for the class records.
First, let `phi_k` be the class record for class `k`,
Let us define
`pi_(i,j)=Prob(phi_k-=i(j))`
`phi_k-=0(2) rArr phi_k=2*phi_(k-1)`
Proof:Let us suppose that `phi_k=2*n` and `phi_(k-1)!=n`
By definition,
`{(phi_k in R_k,rArr sigma_oo(phi_k)=sigma_oo(2*n)=k),(,rArr sigma_oo(n)=k-1),(,rArr n in R_(k-1)):}`
and,
`phi_(k-1)=min(R_(k-1)) rArr phi_(k-1) le n`
as by hypothesis,
`phi_(k-1) != n rArr phi_(k-1) lt n`
thus,
`2*phi_(k-1) lt 2*n = phi_k`
On the other hand
`sigma_oo(2*phi_(k-1))=sigma_oo(phi_(k-1))+1=k rArr 2*phi_(k-1) in R_k`
which is a contradiction because `phi_k=min(R_k) ` ` square`
Let us consider the Total Stopping Time classes near the `k^(th)` and their records.
If `phi_k` is an element of `R_(k,o)`, `phi_(k-1)` can be either an element of `R_(k-1,o)` or `R_(k-1,o-1)`
The completeness being `c=o/k`, the probabilities are respectively:
`{(Prob(phi_(k-1) in R_(k-1,o))=1-c),(Prob(phi_(k-1) in R_(k-1,o-1))=c):}`
If we define `R_(k,o,i)` the set of elements of `R_(k,o)` which are congruent to `i` modulo `3`, and `r_(k,o,i)` the number of elements of this set, the following graph summarizes the origin of the elements of and their quantities.
Any element of `R_(k,o)` can be either the predecessor of an element of `R_(k-1,o)` by an Even Step or the predecessor of an element of `R_(k-1,o-1,2)` by an Odd Step
And as all those sets are disjoints:
`r_(k,o)=r_(k,o,0)+r_(k,o,1)+r_(k,o,2)`
`r_(k,o)=r_(k-1,o)+r_(k-1,o-1,2)`
Per the binomial model:
`r_(k,o)=(((k),(o)))/3^o`
`r_(k-1,o)=(((k-1),(o)))/3^o=(k-o)/k*(((k),(o)))/3^o=(k-o)/k*r_(k,o)`
`rArr Prob(phi_(k-1) in R_(k-1,o))=r_(k-1,o)/r_(k,o)=(k-o)/k=1-c`
`r_(k-1,o-1)=(((k-1),(o-1)))/3^(o-1)=o/k*(((k),(o)))/3^(o-1)=(3*o)/k*r_(k,o)`
and if we consider the elements of `R_(k−1,o−1)` evenly distributed among equivalence classes modulo `3`:
`r_(k-1,o-1,2)=r_(k-1,o-1)/3=o/k*r_(k,o)`
`rArr Prob(phi_(k-1) in R_(k-1,o-1))=r_(k-1,o-1,2)/r_(k,o)=o/k=c`
We can verify that `Prob(phi_(k-1) in R_(k-1,o))+Prob(phi_(k-1) in R_(k-1,o-1))=1 ` `square`
The elements of `R_(k,o)` have successors following a trajectory with `o=c*k` odd steps and `(k-o)=(1-c)*k` even steps.
As a consequence they are coming from `R_(k-1,o-1)` with probability `c` and from `R_(k-1,o)` with probability `(1-c)`.
`{(Prob(F(phi_k) in R_(k-1,o))=(1-c)),(Prob(F(phi_k) in R_(k-1,o-1)=c)):}`
`phi_k` being even is then: `phi_(k-1)` is an element of `R_(k-1,o)` and `phi_k` is coming from `R_(k-1,o)`
This probability is:
`{:(pi_(0,2),=Prob(phi_k-=0(2))),(,=Prob(phi_(k-1) in R_(k-1,o)) xx Prob(F(phi_k) in R_(k-1,o))),(,=(1-c)^2):}`
For class records up to class 1000, the average completeness is 0,58327.
This leads to `pi_(0,2)=0,17366`, or an estimation of 173,66 even class records.
The real number of even class records is 174.
In the Figure bellow we show `pi_(0,2)` and the real number of even class records up to class `k`.
Repartition in Basis 3
Using the properties of Threads (see Threads Chapter) we can estimate that the repartition of class records
in equivalence classes modulo 3, as shown in Figure bellow.
`{:(hat pi_(0,3)=0.3830),(hat pi_(1,3)=0.2604),(hat pi_(2,3)=0.3566):}`
For the 1445 first class records, the real proportions are:
`{:(pi_(0,3)=542//1445=0.3751),(pi_(1,3)=398//1445=0.2754),(pi_(2,3)=505//1445=0.3495):}`
Prime Class Records
Taking into account:
The model for odd Class records: `pi_(1,2)=1-pi_(0,2)=2*c-c^2`,
the estimation for a class record non being divided by 3: `1-pi_(0,3)`,
and the basic probability of a number `n` to be prime: `1/ln(n)`,
The probability for each Class Record `phi_k` having a completeness `c_k` to be a prime is:
`Prob(phi_k text{ is prime})=Prob(phi_k-=1(2)xxProb((phi_k-=1(3)vv(phi_k-=2(3))xx
Prob(phi_k text{ is prime}|not(phi_k-=0(2))^^not(phi_k-=0(3)))`
`Prob(phi_k text{ is prime})=pi_(1,2)*(pi_(1,3)+pi_(2,3))*1/ln(phi_k)*(1/2)^(-1)*(2/3)^(-1)`
`Prob(phi_k text{ is prime})=(3*(2*c_k-c_k^2)*(1-pi_(0,3)))/ln(phi_k)`
And the number of prime estimated up to the class `k` is:
`sum_(i=0)^k((3*(2*c_i-c_i^2)*(1-pi_(0,3)))/ln(phi_i))`
The Figure bellow shows this estimation and the real number of prime class records up to class 1445.
NUMBER OF PRIME FACTORS FOR Class Records
Taking into account:
the estimation of prime factors for any number
`n`, `omega(n)=ln(ln(n))`
The Figure bellow shows this estimation and the real number of prime
factors for class records up to class 1445.
