Problems & Puzzles: Conjectures

Conjecture 86.  The Majid's Conjecture

Majid Azimi wrote on October 3, 2020:

Recently I came up with the following conjecture, not studied previously, as far as I know:

Conjecture:

"There is at least one couple of twin primes between the square of consecutive odd number, (2n-1)^2 and (2n+1)^2, for n=>1"

I have tested for n<=10^9 and the conjecture holds. See here all the data I have computed, shown by ranges up to n<=10^6.

Two examples:

 n 2n-1 2n+1 (2n-1)^2 (2n+1)^2 Quantity of couples  of twin primes 1 1 3 1 9 2; (3,5), (5,7) 2 3 5 9 25 2; (11,13), (17,19)

Not a proof of my conjecture, but just a graphical description of the behavior of the Count of couples of twin primes (Q) between (2n-1)^2 and (2n+1)^2, I offer you the following graph of data:

Fig A: Count of couples of twin primes between the square of consecutive odd integers
(
Vertical line is count of couple twins between (2n-1)^2 and (2n+1)^2 and horizontal line is n, for n=>1)

Q1. Can you tell if this Conjecture has been proposed and/or studied before by others?

Q2. Can you find a counterexample, or prove this conjecture.

On november 14, 2020, Alain Rochelli wrote:

I send you a contribution concerning the conjecture 86.

Still today, in number theory, a main problem is to ascertain whether there exist infinitely many twin primes.

However, based on heuristic considerations about the distribution of twin primes, an approximation of Q(x) - denoted as the number of couples of twin primes < x - seems to be :

Q(x) = 1,32*x / (logx)^2

This asymptotic formula (for x to infinite) is in agreement with the numerical results given by Majid.

For example :

n = 10 000

(2n-1)^2 = 399 960 001 = A and (2n+1)^2 = 400 040 001 = B

Q(B) - Q(A) = 1 345 974 - 1 345 732 = 242

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On November 30, 2020 Némo Sauvion wrote:

I send you a remark about Majid's conjecture. It seems to be close to Legendre's conjecture, which states that there is a prime number between n^2 and (n+1)^2 for every positive integer n.

I found on the OEIS two sequences about a similar problem, closer to Legendre's conjecture and which can be expressed as follows :

For every positive integer n except 1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77 and 122, there is at least one couple of twin primes between n^2 and (n+1)^2. (It was tested for n up to 10^7).

A091592 (https://oeis.org/A091592) is the list of known exceptions and A091591 (https://oeis.org/A091591) gives numbers of pairs of twin primes between n^2 and (n+1)^2.

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Alexei Kourbatov wrote on Dec 20, 2020:

To complement earlier remarks by Alain Rochelli and Némo Sauvion: a rigorous proof of Conjecture 86 seems beyond reach, but estimates based on the Hardy-Littlewood k-tuple conjecture give a reasonable prediction of the growing number of twins between squares.

A number of very similar conjectures have been studied in my paper

"Maximal gaps between prime k-tuples: a statistical approach"

published in J. Integer Sequences, vol. 16 (2013), Article 13.5.2;

Here is an excerpt from Section 7 of the above paper:
The following conjectures generalize Legendre's conjecture about primes between squares:

- For each integer n>0, there is always a prime between n^2 and  (n+1)^2. (Legendre)

- For each integer n>122, there are twin primes between n^2 and  (n+1)^2. (A091592)

- For each integer n>3113, there is a prime triplet between n^2 and  (n+1)^2.

- For each integer n>719377, there is a prime quadruplet between n^2 and (n+1)^2.

- For each integer n>15467683, there is a prime quintuplet between n^2 and  (n+1)^2.

- There exists a sequence {s_k} such that, for each integer n > s_k, there is a prime k-tuplet between n^2 and (n+1)^2.
(This {s_k} is OEIS A192870: 0,122,3113,719377,...)

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