The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem

Abstract: We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+η$, $μ^2(η)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length unlike previous versions.

• The paper establishes Goldbach-type representation results for sums involving both primes and reversed primes across various bases.
• It refines the Zsiflaw–Legeis theorem by providing effective asymptotics for reversed primes in arithmetic progressions using advanced exponential sum estimates.
• It adapts the Hardy–Littlewood circle method to overcome digital irregularities, offering insights for computational number theory and cryptographic applications.

Summary of "The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem" (2605.21876)

Introduction and Framing

This paper extends additive number theory into the domain of reversed primes, that is, primes written backwards in a fixed base $b \geq 2$. The authors introduce and resolve several variants of Goldbach-type conjectures concerning representations of integers as sums involving primes and reversed primes, especially in nonstandard bases. Central to the analysis is a refined distributional theorem for reversed primes in arithmetic progressions (the Zsiflaw--Legeis theorem), together with adaptations of the Hardy–Littlewood circle method.

Main Results

Goldbach-Type Assertions for Reversed Primes

The paper establishes multiple strong additive results:

• Every sufficiently large odd integer is expressible as the sum of either a prime and two reversed primes ($N = p_1 + \#p_2 + \#p_3$), or two primes and a reversed prime ($N = p_1 + p_2 + \#p_3$).
• Almost all even integers are the sum of a prime and a reversed prime ($N = p_1 + \#p_2$), with an explicit exceptional set bound of size $O_{b,A}(x/(\log x)^A)$.
• Every sufficiently large integer is the sum of a reversed prime and a square-free number ($N = \#p + \eta$ with $\mu^2(\eta) = 1$).

Each theorem admits an asymptotic formulation, given in terms of logarithmic weights (for analytical tractability), and includes singular series factors quantifying local obstructions arising from digital properties and uniformity in arithmetic progressions.

Refined Zsiflaw–Legeis Theorem

Traditionally, distributional results for reversed primes in residue classes require fixing the digital length. The authors exhibit:

• An effective asymptotic for the counting function of reversed primes in arithmetic progressions, without fixing digit length, for all bases $b \geq 2$.
• Enhanced range: for moduli $q$ up to $\exp(c\sqrt{L})$ (with $N = p_1 + \#p_2 + \#p_3$0 digit length), and with error term $N = p_1 + \#p_2 + \#p_3$1, paralleling and sometimes outperforming classical Siegel–Walfisz bounds.

This generalization is critical for applications within the circle method, as it allows uniform estimates when reversed primes are aggregated by value rather than digit length, and permits logarithmic weights.

The theorem holds for reversed primes coprime to $N = p_1 + \#p_2 + \#p_3$2, a technical condition that avoids digit-related periodicities and large gaps in digital reverses, which are severe for composite bases.

Circle Method Application

Employing the refined Zsiflaw–Legeis results, the authors adapt the circle method to handle sums involving reversed primes. For ternary and binary additive problems (three and two summands), asymptotics are obtained with explicit singular series and combinatorial main terms. The major arc estimates follow classical analytic number theory procedures, adjusted for digital structure, while minor arcs rely on bounds for exponential sums over reversed primes, supported by a suite of digital function techniques.

Limits and Conjectures

Several conjectures are articulated:

• Hcabdlog's conjecture: For suitable coprimality restrictions, every sufficiently large even $N = p_1 + \#p_2 + \#p_3$3 is representable as the sum of a prime and a reversed prime ($N = p_1 + \#p_2 + \#p_3$4).
• Conjectures on sums purely of reversed primes: In certain bases (notably prime bases), every sufficiently large even $N = p_1 + \#p_2 + \#p_3$5 should be the sum of two reversed primes; for $N = p_1 + \#p_2 + \#p_3$6, every sufficiently large odd $N = p_1 + \#p_2 + \#p_3$7 should be the sum of three reversed primes.

It is proven and discussed that such representations cannot always hold in composite bases due to digital gaps.

Numerical and Singular Series Considerations

Explicit bounds are provided for main terms, e.g., $N = p_1 + \#p_2 + \#p_3$8, guaranteeing the validity of the asymptotics. The singular series quantify the density reduction arising from digital coprimality, analogously to local factors in classical Goldbach–Vinogradov problems.

Methods and Technical Innovations

Weakly Digital Functions

The paper leverages the framework of weakly digital functions (introduced in previous works) to encode congruence and reverse-digit conditions within exponential sums and to exploit uniformity across residue classes. This addresses the digital wildness that complicates the distribution of reverses, especially for composite bases.

Exponential Sum Estimates

Bounds akin to Vinogradov's classical estimate for exponential sums over primes are generalized to reverses, permitting minor arc control in the analytic framework. For purely reversed-prime problems, a conditional hypothesis (“reverse Vinogradov”) is proposed, indicating the necessity of further progress in digital exponential sum estimates for binary problems.

Discussion and Implications

Theoretical Implications

These results bridge digital number theory and additive theory, demonstrating that reversed primes, despite their pronounced irregularity and gaps, exhibit sufficient uniformity for Goldbach-type representations, at least when primes and reverses are combined. The refined distribution in arithmetic progressions underpins additive applications across bases.

Practical Implications

Applications concern both computational number theory (for representing integers with digital constraints) and the theory of cryptographic constructions where digital reverses are relevant. The bounds on exceptional sets and constructive asymptotics allow explicit computation of representable integers for large $N = p_1 + \#p_2 + \#p_3$9.

Extension and Future Work

The “reverse Goldbach” and “reverse Schnirelmann” constants are introduced, with analysis showing their growth with base complexity: the minimum number of reversed primes needed for universal representation can be arbitrarily large for certain bases. Further, conditional asymptotics for pure reversed-prime sums depend on progress in digital exponential sum estimates.

It is anticipated that future developments will lead to improved minor arc bounds for reverses, potentially paralleling classical exponential sum techniques, and that extension to higher arity additive problems with digital constraints will follow similar frameworks. Work on distributions within composite vs. prime bases and further singular series analysis are pressing.

Conclusion

This paper establishes deep connections between additive number theory and the digital properties of integers, providing both effective and asymptotic results for integer representations involving reversed primes. The refined Zsiflaw–Legeis theorem gives essential distributional input for analytic techniques, and the suite of Goldbach-type results, both proven and conjectural, opens further avenues for research into digital additive phenomena and exponential sum estimates for highly structured sets.