Alain Rocchelli's
conjecture 106 can be derived from a simple random model of prime
numbers.

Numerical computations
involving primes up to 7*10^13 are consisted with it.

Let's introduce one
more mathematical sequence E(n) along with N(n) and D(n):

E(1)=1;

for n>1, E(n)=a(n)=prime(n+1)-prime(n)
if a(n)==a(n-1) and E(n)=0 otherwise.

For n=1, equality
N(1)+D(1)+E(1)=3=prime(2) holds;

for n>1, equality
N(n)+D(n)+E(n)=a(n)=prime(n+1)-prime(n) holds;

then equality
Sum_N(n)+Sum_D(n)+Sum_E(n)=prime(n+1) holds for every n by
induction on n.

Set counting
interpretation.

Positive integers can
be partitioned into three sets Set_N, Set_D, Set_E as follows.

Arbitrarily assign 1 to
Set_D, 2 to Set_E, 3 to Set_N.

For n>1, consecutive
primes prime(n) and prime(n+1) uniquely determine a "gap-interval"
I(n) with even length:

I(n)={prime(n)+1<=k<=prime(n+1)}.

Assign all integers k
from a given gap-interval I(n) to the same set:

to Set_N if a(n)>a(n-1),

to Set_D if a(n)<a(n-1),

to Set_E if a(n)==a(n-1).

From these definitions,
cumulative sums Sum_N(n), Sum_D(n), Sum_E(n) can be viewed as
counting functions.

Considering integers k
between 1 and prime(n+1) included, there are:

Sum_N(n) integers
belonging to Set_N,

Sum_D(n) integers
belonging to Set_D,

Sum_E(n) integers
belonging to Set_E.

Asymptotic conjectured
behaviour.

As n diverges to
infinity:

ratio R_N(n)=Sum_N(n)/prime(n+1)
converges to 0.75,

ratio R_D(n)=Sum_D(n)/prime(n+1)
converges to 0.25,

ratio R_E(n)=Sum_E(n)/prime(n+1)
converges to 0,

ratio R_N(n)/R_D(n)=Sum_N(n)/Sum_D(n)
converges to 0.75/0.25 = 3.

Random model of prime
numbers (modified Cramer's model).

Integers from 1 to 7
are prime with probability either 0 or 1, according to their actual
primality status.

An integer k > 7 is
prime with probability q(k)=0 if k is even and q(k)=2/log(k) if k is
odd.

Given any two positive
integers, their primality probabilities are independent.

For large x, the
expected number of primes between 1 and x is asymptotic to x/log(x),
consistently with Prime Number Theorem.

Let P_N(k) be the
probability of integer k belonging to Set_N.

Its cumulative sum from
k=1 to k=prime(n+1) will be an estimate of Sum_N(n).

Its average value over
the same interval will be an estimate of R_N(n).

Similarly define P_D(k)
and P_E(k).

For large x,
approximations to P_N(x), P_D(x) and P_E(x) will be computed as
follows.

If k is odd and close
to x, then q(k) is close to constant Q = 2/log(x).

Integer x belongs to a
gap-interval with length 2*t, preceded by a gap-interval with length
2*u, with probability close to:

t*Q*(1-Q)^(t-1)*Q*(1-Q)^(u-1)*Q.

Taking suitable
summations for t and u both growing from 1 to infinity, I obtained
the following results:

P_E(x) ~= Q/(2-Q)^2

P_D(x) ~= (1-Q)/(2-Q)^2

P_N(x) ~=
(3-Q)*(1-Q)/(2-Q)^2

As x diverges to
infinity:

Q converges to 0;

P_E(x) converges to 0
too;

P_D(x) converges to
1/4;

P_N(x) converges to
3/4.

Numerical computations.

n = 2269432871305;

prime(n) =
69999999999971;

prime(n+1) =
70000000000009;

Sum_N(n) =
50879234052519;

Sum_D(n) =
18342798190645;

Sum_E(n) =
777967756845;

R_N(n) = 0.72684620075;

R_D(n) = 0.26203997415;

R_E(n) =
0.011113825098;

Sum_N(n)/Sum_D(n) =
2.77379893317.

Random model
predictions.

x = 70000000000009;

Q = 0.062736208967;

P_N(x) ~=
0.73354585688;

P_D(x) ~=
0.24973782032;