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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
***
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).
***
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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