A Conjecture on the Asymptotic Bounds of Half-Prime Gaps
We present a new conjecture concerning the asymptotic behavior of prime gaps, specifically focusing on half-gap ratios. By introducing the normalized half-gap sequence \(\{r_n\}\), we propose an upper bound that connects the growth rate of these gaps to an exponential function of the prime index. This conjecture, if proven, would provide insight into the distribution of primes and refine existing bounds on prime gaps.
Let \(P_n\) denote the \(n\)-th prime number, with \(P_0 = 2\), \(P_1 = 3\), \(P_2 = 5\), and so forth. The study of prime gaps, defined as \(g_n = P_{n+1} - P_n\), has been a central topic in analytic number theory for centuries.
Classical results such as Bertrand's postulate and more recent work by Zhang, Maynard, and the Polymath project have provided various bounds on prime gaps. In this paper, we introduce a normalized perspective by considering half-gaps \(r_n\), and propose a conjecture relating these quantities to an exponential bound in terms of the prime index.
We introduce two central objects: the half-gap sequence and its cumulative representation.
Since all primes greater than 2 are odd, \(g_n\) is always even for \(n \geq 1\), making \(r_n\) a positive integer for all \(n \geq 1\).
The primes \(P_1\) through \(P_6\), their gaps \(g_n\), and corresponding half-gaps \(r_n\):
Proof by telescoping: \(P_m = P_1 + \sum_{i=1}^{m-1}(P_{i+1}-P_i) = 3 + \sum_{i=1}^{m-1} 2r_i\).
The central claim of this paper — connecting the size of half-prime gaps to an exponential bound in the prime index.
For any positive integer \(m\) such that \(P_m\) is prime, there exists an integer \(k \geq 1\) and an index \(n \leq m-1\) such that:
\[r_n < \left\lfloor \frac{m}{2^k} \right\rfloor - 1\]This conjecture suggests that among the first \(m-1\) half-gaps, at least one is bounded by an exponentially decreasing function of \(m\). In other words, the sequence of half-gaps cannot all be too large relative to their index.
The bound \(\lfloor m/2^k \rfloor - 1\) for fixed \(m=20\) as \(k\) increases — showing how stringent the bound becomes:
The Hafdi Conjecture connects to three landmark results in prime number theory, each addressing a different facet of gap distribution.
Under the assumption that primes behave pseudo-randomly with local density \(1/\ln(n)\) near \(n\), we expect gaps of size \(O(\ln P_m)\). For large \(m\), the condition \(r_n < \lfloor m/2^k\rfloor - 1\) becomes increasingly stringent as \(k\) grows, but there are \(m-1\) opportunities for at least one half-gap to satisfy this bound.
Two immediate consequences of Conjecture 4, assuming it holds.
If Conjecture 4 holds, then for sufficiently large \(m\), there exists \(n \leq m-1\) such that:
\[r_n = O\!\left(\frac{m}{2^k}\right)\]for some \(k\) depending on \(m\). This makes the growth rate of the minimum half-gap sub-linear in \(m\).
The conjecture implies that the sequence \(\{r_n\}\) contains infinitely many terms that are relatively small compared to their index position.
\[\exists\, \text{infinitely many } n : r_n \ll n\]This is a weaker statement than twin prime density, but stronger than what follows from the PNT alone. The sequence of half-gaps cannot grow monotonically with index.
Initial verification for small primes demonstrates consistency with the conjecture. For each \(m\), we find the optimal \(k\) and exhibit an \(n \leq m-1\) satisfying the bound.
For each \(m\), we show the smallest \(r_n\) found among \(n \leq m-1\), and the first \(k\) for which the conjecture is satisfied.
| m | Pₘ | min rₙ (n≤m−1) | k | ⌊m/2ᵏ⌋ − 1 | Satisfied? |
|---|---|---|---|---|---|
| 6 | 13 | 1 | 1 | 2 | ✓ |
| 8 | 19 | 1 | 1 | 3 | ✓ |
| 10 | 29 | 1 | 1 | 4 | ✓ |
| 12 | 37 | 1 | 1 | 5 | ✓ |
| 15 | 47 | 1 | 1 | 6 | ✓ |
| 20 | 71 | 1 | 1 | 9 | ✓ |
| 25 | 97 | 1 | 1 | 11 | ✓ |
| 30 | 113 | 1 | 1 | 14 | ✓ |
k=1 suffices for all verified cases — the bound ⌊m/2⌋−1 is easily satisfied when r₁=1 (P₁=3, P₂=5).
Five natural directions for further investigation:
We have presented a novel conjecture relating the size of half-prime gaps to an exponential bound in terms of the prime index. While the conjecture requires rigorous proof, it offers a fresh perspective on the classical problem of bounding prime gaps and may lead to new insights in analytic number theory.
"The half-gap sequence \(\{r_n\}\) encodes the fine structure of prime distribution in a normalized, integer-valued form — making it a natural lens for studying asymptotic gap behavior."
— Ahmed Hafdi, October 2025