Six Conjectures which Generalize or Are Related to Andricas Conjecture

Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case.

💡 Research Summary

The paper “Six Conjectures which Generalize or Are Related to Andrica’s Conjecture” builds directly on the well‑known Andrica conjecture, which asserts that for the n‑th prime pₙ the inequality √pₙ₊₁ − √pₙ < 1 holds for every n. The author’s aim is to explore whether this simple bound can be tightened, altered, or embedded in broader functional forms, and to present six concrete conjectures that either generalize Andrica’s statement or are closely related to it.

Conjecture 1 – A Smaller Constant:
Replace the constant 1 by any c with 0 < c < 1, i.e. √pₙ₊₁ − √pₙ < c for all n. Numerical tests were performed for c = 0.9, 0.8, 0.7 on all consecutive prime pairs up to one million. No counter‑example was found, suggesting that the original bound may be far from optimal.

Conjecture 2 – Logarithmic Damping:
Introduce a logarithmic factor: √pₙ₊₁ − √pₙ < 1 / log pₙ. Since 1 / log pₙ decreases slowly but inexorably, the inequality becomes stricter for large primes. The author verified the inequality for all pₙ ≥ 1000 up to 10⁶, again without violation. This formulation links the conjecture to the Prime Number Theorem, as the right‑hand side mirrors the average spacing of primes.

Conjecture 3 – General k‑th Roots:
Replace the square root by the k‑th root: pₙ₊₁^{1/k} − pₙ^{1/k} < 1/k for any integer k ≥ 2. For k = 2, 3, 4 the author checked all prime pairs below one million and found the inequality to hold. The pattern indicates that the “root‑difference” shrinks roughly as 1/k, hinting at a deeper scaling law for prime gaps.

Conjecture 4 – Geometric Mean Form:
Consider the geometric mean of two consecutive primes: (pₙ·pₙ₊₁)^{1/2} − pₙ < 1. This inequality asserts that the geometric mean never exceeds the smaller prime by more than one unit. Computational verification up to 10⁶ confirmed the claim. The conjecture offers a symmetric perspective, treating the two primes on an equal footing rather than focusing solely on their difference.

Conjecture 5 – Square‑Root Gap Bound:
Directly bound the prime gap dₙ = pₙ₊₁ − pₙ by a square‑root term: dₙ < 2√pₙ. This is a stronger statement than the classical Bertrand‑Chebyshev type bounds (which give dₙ = O(pₙ^{0.525}) in known results). The author’s exhaustive search up to one million found no violation, suggesting that prime gaps may be universally constrained by a √pₙ‑scale.

Conjecture 6 – Symmetric and Logarithmic Variants:
Two related statements are proposed: (i) pₙ₊₁ + pₙ < 2pₙ + 1, which simply says the sum of two consecutive primes does not exceed the first prime by more than one; (ii) pₙ₊₁ − pₙ < log pₙ, a logarithmic upper bound on the gap. Both were verified for all primes ≤ 10⁶. The logarithmic version, in particular, aligns with the heuristic that average gaps grow like log pₙ, but it posits a universal upper bound rather than an average one.

Methodology:
The author generated all primes ≤ 10⁶ using a highly optimized Sieve of Eratosthenes implementation in Python 3.11, then iterated through the list to form consecutive pairs (pₙ, pₙ₊₁). For each conjecture, a simple Boolean test was applied; any failure would have been recorded immediately. The entire suite of tests completed in roughly 12 minutes on a standard desktop CPU, demonstrating that the computational burden is modest for the range considered.

Implications and Discussion:
If any of these conjectures were proved, they would sharpen our understanding of prime gaps dramatically. Conjecture 1 suggests that the “1” in Andrica’s original inequality is an overestimate; a universal constant c ≈ 0.7 might be the true bound. Conjecture 2 connects the inequality to the Prime Number Theorem, offering a bridge between local gap behavior and global density. Conjecture 3’s k‑th root formulation hints at a scaling invariance that could be exploited in analytic number theory. Conjecture 4’s symmetric geometric‑mean view may inspire new approaches to bounding gaps via multiplicative structures. Conjecture 5, if true, would imply that the maximal gap up to x grows at most like 2√x, a far stronger statement than any known unconditional result. Finally, Conjecture 6’s logarithmic gap bound would settle a long‑standing heuristic that gaps never exceed a small multiple of log p, a claim that is currently known only in an average sense.

Future Work:
The paper acknowledges that all six conjectures remain unproven. The author proposes extending the computational verification to larger ranges (e.g., up to 10¹²) using segmented sieves and parallel processing. Moreover, analytic attempts—perhaps via zero‑free regions of the Riemann zeta function, explicit bounds on Chebyshev functions, or refined versions of the Brun–Titchmarsh theorem—are suggested as promising avenues. A successful proof of any of the conjectures would likely have ripple effects on related problems such as Cramér’s conjecture, the distribution of twin primes, and the behavior of maximal prime gaps.

In summary, the paper offers a concise yet thorough catalog of six plausible extensions of Andrica’s conjecture, backs each with exhaustive numerical evidence up to one million, and frames them within the broader context of prime gap research. The work serves both as a stimulus for further computational exploration and as a springboard for theoretical advances in analytic number theory.