Firoozbakht’s conjecture

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In this article, we examine the relationship between Farideh Firoozbakht's famous conjecture about prime numbers, which states that the sequence 1-9 where 2-5 is the nth prime number, is strictly decreasing, and the prime gap function 3-5 which is the difference between the nth and (n + 1)th prime numbers. Furthermore, Firoozbakht's conjecture is linked to upper limits for the prime gap function in terms of n. Asymptotic outcomes for both the conjecture and the function help us understand the interrelationships.

Farideh Firoozbakht's initial conjecture in 1982 was that the sequence 4-6 is strictly decreasing, i.e.,
5-3
and the first few values of 6-6 are shown below:

7-5

and the speculation has been confirmed for all primes less than 4x(10)^18.

Related forms of Firoozbakht’s conjecture

Following are some alternative formulations of the hypothesis. Here n∈N.

  • 8-4 (1),
  • 9-3 (2),
  • 10-1 (3),
  • 11-2 (4)

These are easily provable by direct analysis and are by no means a complete list. The left hand side of (2), however, is the nth prime gap 3-6, and if the conjecture is correct, 12-1 is an upper bound in terms of n for 13.

In fact, if (1) is accurate, an upper bound of 13-1 is found to be 14. (1) may also be used to improve on the Baker-Harman-Pintz's upper bound of 15 for 16. More conjectural upper limits for 13-3 in terms of n will be showed, but for the time being, we will focus on certain known findings that give an approach to demonstrating (1).

Evidence to back up the conjecture

Proving (1) for infinitely many n∈N:

Zhang demonstrated that,
17
Outcome: There exists an infinite number of n∈N such that 18
Proof: We assume 19 and 20
Then, according to Zhang's Theorem, T is infinite and
21-1
Now, 22
Specifically, for any sufficiently big n∈T, 30. This is identical to the statement of the theorem. Thus,
Zhang’s Theorem ⇒ (1) applies for infinitely many n ∈ N

Proving (1) for almost all n ∈ N:

In order to prove this statement, we will require the following two well-known theorems.
Theorem 1: Let us consider the set of the first n−1 differences between consecutive primes, i.e 23. Let 24 be the number of differences in this set such that 25-1 where ϵ > 0 is fixed. We then have the limit 26. Therefore, if 27 is the number of differences in this set such that 28 we have that 29. That is, practically all of the set's differences 31 fulfil 32
Theorem 2: The following asymptotic formula holds
33

Proof: We have 34. The prime number theorem 35, consequently 36, derives directly from the formula exp(x)=1+x+o(x)(x→0). The theorem is proved.
Outcome: The inequalities in (1) to (4) hold for practically all successive prime pairings 37 and 2-6.

Proof: According to Theorem 2, since 38 is asymptotically comparable to 39, there must be some value of n∈N, such that ∀j>n, j∈N, 40.
Furthermore, theorem 1 with ϵ=12 shows that for nearly all pairs of successive primes 41, the inequality 42 holds for almost all pairings of consecutive primes 41-1. This is inequality (2), and because (2) and (1) are similar, (1) holds true for almost all n∈N.

Upper boundaries of the prime gap function

The prime gap function, 3-7, is inextricably linked to Firoozbakht's conjecture. There are a few speculative upper bounds on 13-4.
It is proved that 43
It can also be proven that 44

Following the demonstration of the previous implication, we shall show how (1) follows from the assumed reality of another such conjectural upper bound, one of a family of such bounds.
Outcome: 45
Proof: We notice that (1) has been proven for all primes less than 4x(10)^18, which is why we include it in the stated condition.
Using the Panaitopol formula for π(x), we get an upper bound for the prime-counting function π(x), which is provided by:
47
We are not including our x ≥ 9.25 constraint since we are examining primes higher than 4x(10)^18, and primes up to the limit have already been proven to obey the conjecture. As a result, the primes in question are large enough.
We know using calculus,
48
Thus we get from the given inequality,
49
From (i), (ii) and (iii), we get,
50
which is (4), one of the equivalent forms of (1). As a result, we have demonstrated that if the higher bound indicated above holds, then (1) is true.
There are two additional conjectured upper boundaries for 13-5, which are as follows:
51
and the truth of either of these two implies (1).

Program to verify the conjecture

Following is a C++ Program to verify the conjecture:

#include <bits/stdc++.h>
using namespace std;

void firoozbakhtConjecture(int n)
{
    bool prime[n + 1];
    memset(prime, true, sizeof(prime));
    for (int p = 2; p * p <= n; p++)
    {
        if (prime[p] == true)
        {
            for (int i = p * p; i <= n; i += p)
            prime[i] = false;
        }
    }
    double i=1;
    for (int p = 2; p <= n; p++)
        if (prime[p])
        {
            cout << double( pow(p,(1/i)) ) <<"\n";
            i++;
        }
}

int main()
{
    int n = 100;
    firoozbakhtConjecture(n);
    return 0;
}

The above C++ program will yield the following output:
Untitled

The series keeps on decreasing for a large range and hence the conjecture can be proved till that given range. You can run it continuously till two consequtive elements increase (which will disprove the conjecture).

These techniques are used to verify conjectures till a given limit.

Associated conjectures

Three more conjectures were proposed in relation to the Firoozbakht Conjecture.
The Nicholson Conjecture-
52-1

Forgues' Conjecture-
53-2

Farhadian Conjecture-
54

To see how these conjectures connect to the Firoozbakht Conjecture, consider the following inequalities:
Theorem 1:
55
This inequality is know as the Rosser’s Theorem.

Theorem 2: Farhadian ⇒ Nicholson ⇒ Firoozbakht ⇒ Forgues
Proof Assume the Farhadian Conjecture is correct. Thereafter, using logarithms,
56
From previous theorem,
57
This suggests the Nicholson Conjecture. Assuming the Nicholson Conjecture, Rosser's Theorem states that
58
which is the same as
59
This corresponds to the Firoozbakht Conjecture. Taking logarithms out from previous inequality, we get
60
When we increase to the n-power, we obtain
61
which is the Forgues Conjecture.

The Firoozbakht conjecture is one of the most powerful upper limits on prime gaps, rivalling the Cramér and Shanks conjectures. Cramér predicted that the gaps around p are no larger than 62, while Shanks proposed the asymptotic equivalence 63 for maximal prime gaps g.