Long strings of composite values of polynomials and a basis of order 2

Abstract: We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}={n_1-m,\ldots, n_1+m}$ and $I_{2}={n_2-m, \ldots, n_2+m}$, where $m = [(\log N) (\log \log N)^{1/325525}]$, such that $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and $f(n)$ is composite for any $n\in I_{1}\cup I_{2}$. This extends the result in [5] which showed the same result but with $f(n)=n$.

Long Strings of Composite Values of Polynomials and a Basis of Order 2

Artyom Radomskii's paper, "Long strings of composite values of polynomials and a basis of order 2," presents a notable extension to previous work by Gabdullin and Radomskii, wherein they demonstrated that strings of consecutive integers could all yield composite values when plugged into polynomial functions. Specifically, this paper focuses on polynomials that are irreducible over $\mathbb{Q}$, have positive leading coefficients, and explores the implications on integers that can avoid prime numbers entirely in the values of these polynomials.

Main Contributions

Radomskii achieves a significant generalization: given a polynomial $f: \mathbb{Z} \to \mathbb{Z}$ with certain properties, there exist two strings of consecutive positive integers, $I_1$ and $I_2$, within the bounds $[1, N]$ such that $f(n)$ is composite for any integer $n$ within the union $I_1 \cup I_2$. Theoretical underpinnings are explored to prove that the composite integer density is ample, primarily using advanced sieve methods.

This work builds upon prior results involving linear functions, such as $f(n) = n$, enhancing the theoretical framework to encompass non-linear and irreducible polynomials. By modifying existing constructions and introducing analytical techniques, Radomskii extends the results to a broader class of functions.

Technical Insights

1. Density Constants & Bounds: Radomskii defines $C(\rho)$, a supremum for real numbers $\delta$, used to establish bounds that ensure the string lengths, $m$, defined mathematically as $[\log(N) (\log\log(N))^\delta]$, are sufficiently large.
2. Probabilistic Methodology: The paper extensively employs probabilistic techniques to manage residue classes and compute expectations over composite numbers. Essential probabilistic tools include estimating expectations and variances, ensuring that valid outcomes can form with high likelihood.
3. Numerical Results & Assumptions: Through detailed setups and lemmas, Radomskii establishes various inequalities and numerical thresholds necessary to assure high probative results. Constants derived play a crucial role in balancing technical constraints and general applicability.
4. Conjecture Context: This work operates within a framework framed by open conjectures like those proposed by Bouniakowsky and hypotheses by Bateman and Horn regarding the distribution and density of prime numbers in polynomial sets.

Implications and Future Directions

Radomskii’s approach of forming strings of composite values in polynomials widens the potential for practical applications, such as constructing sets of integers with specific non-prime characteristics. The implications extend to number theory, particularly in understanding the distribution of composite numbers and refining sieve methods that might further induce new conjectures about polynomial behavior.

Future research can explore:

• Exploring the constraints and variations of $\delta$ to uncover new polynomial classes.
• Developing more effective computational techniques for managing high-order polynomials.
• Applying this theoretical advancement to cryptographic systems where prime avoidance is beneficial.

Conclusion

The paper offers a methodical expansion to the understanding of polynomials generating composite values over extended integer intervals. Radomskii's contribution is robust in its mathematical formulations and extends emerging sieve applications in prime number theory, providing fertile ground for future innovations and explorations in algebraic and analytic number theory.