Problems & Puzzles: Conjectures

Conjecture 106. Conjecture about gaps between consecutive primes.

On June 11, 2024 Alain Rocchelli sent the following conjecture:

A001223 is related to the gaps between consecutive primes, i.e. :

a(n) = prime(n+1) – prime(n) with the two first terms a(1)=3-2=1 and a(2)=5-3=2

Let be {N(n)} and {D(n)} as the two mathematic series defined by:

N(1)=1 ; D(1)=1

For n>1 ; N(n)=a(n)= prime(n+1) – prime(n) if a(n)>a(n-1) and N(n)=0 if a(n)<=a(n-1)

D(n)=a(n)= prime(n+1) – prime(n) if a(n)<a(n-1) and D(n)=0 if a(n)>=a(n-1)

Also we get for the sums of first terms :

Prime  :   2 ,  3 , 5 ,  7 , 11 , 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

a(n) -->   1 ,  2 ,  2 ,  4 ,  2 ,  4 ,  2 ,  4 ,  6 ,  2 ,  6 ,  4 ,  2 ,  4 ,  6 ,  6

N(n) -->  1 + 2 + 0 + 4 + 0 + 4 + 0 + 4 + 6 + 0 + 6 + 0 + 0 + 4 + 6 + 0

D(n) -->  1 + 0 + 0 + 0 + 2 + 0 + 2 + 0 + 0 + 2 + 0 + 4 + 2 + 0 + 0 + 0

We conjecture that the ratio Sum_N(n) / Sum_D(n) is asymptotic to the value 3 for n to infinity.

Q2. Can you get an explanation ?

Contribution came from Oscar Volpatti on June 21, 2024:

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;
P_E(x) ~= 0.01671632280.

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