Geometric Insights into the Goldbach Conjecture

1. Introduction

The Goldbach conjecture, proposed in 1742, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers [1]. Despite centuries of effort and computational verification up to $4\times {10}^{18}$ [2], the conjecture remains unproven. In this paper, we develop a geometric and combinatorial framework for studying a natural variant: every even integer $2N\ge 8$ is the sum of two distinct primes.

Our variant requires the two primes to be distinct, thus excluding the trivial representations $4=2+2$ and $6=3+3$. This restriction is not merely technical—it emerges naturally from a geometric reformulation that provides structural insight. Specifically, we show that for $N\ge 4$ (so $2N\ge 8$), finding a Goldbach partition with distinct primes is equivalent to finding nested squares whose L-shaped difference region has a semiprime area.

The framework proceeds in three stages.

Stage 1: Geometric equivalence (Section 3). We establish that the variant Goldbach conjecture is equivalent to the statement: for every $N\ge 4$, there exists $M\in [1,N-3]$ such that $N^2-M^2=P·Q$ where $P=N-M$ and $Q=N+M$ are both prime. This reformulation recasts the additive problem as a set intersection problem between a candidate set $C_N$ and a valid set $D_N$, both subsets of $\{1,\dots ,N-3\}$.

Stage 2: Density analysis (Section 4). We prove a tight lower bound on $|D_N|$. The key observation is that a single prime $Q^{*}$ in the interval $(N,2N)$, paired with every odd prime $P<N$, generates exactly $\pi (N-1)-1$ distinct elements of $D_N$. Since the candidate set satisfies $|C_N|=\pi (N-1)-1$, this yields

$$|D_N|\ge \pi (N-1)-1=|C_N|.$$

We further show, using Dusart’s thesis [3], that $(N,2N)$ contains at least ${ln}^{2}N$ primes for $N\ge 3275$, each contributing additional values to $D_N$.

Stage 3: Reduction to intersection (Section 4). The Goldbach conjecture for $2N$ is equivalent to $C_N\cap D_N\ne \varnothing$. Our size bounds establish $|C_N|+|D_N|\ge 2(\pi (N-1)-1)$. For the pigeonhole principle to force a non-empty intersection, one would need $|C_N|+|D_N|>N-3$, which requires a strictly stronger bound on $|D_N|$. We characterise precisely what remains open: one must show either that $|D_N|>(N-3)-\pi (N-1)+1$, or that the structure (not merely the size) of $C_N$ and $D_N$ forces an intersection.

Stage 4: Conditional argument and computational evidence (Section 5 and Section 6). To bridge the gap identified in Stage 3, we introduce the gap function $G\left(N\right)={log}^2\left(2N\right)-((N-3)-|D_N\left|\right)$ and formulate the Density Hypothesis (Hypothesis 1): $G\left(N\right)>0$ for all $N\ge 3275$. If the hypothesis holds, then $|D_N|>(N-3)-{log}^2\left(2N\right)$, and since $|C_N|=\pi (N-1)-1>{log}^2\left(2N\right)$ for large N, the pigeonhole principle forces $C_N\cap D_N\ne \varnothing$, yielding the conjecture. We support the hypothesis with extensive computation: $G\left(N\right)>0$ for all $N\in [4,2^{14}]$, with minima strictly increasing across dyadic intervals.

The remainder of this paper is organised as follows. Section 2 collects the analytic number theory prerequisites from Dusart’s thesis and states Bertrand’s postulate. Section 3 develops the geometric framework, reformulates the Goldbach variant as a set intersection problem, and introduces the key sets $C_N$ and $D_N$. Section 4 contains the unconditional density analysis and main structural results. Section 5 presents computational evidence. Section 6 develops the conditional argument via the Density Hypothesis. Section 7 discusses significance and future directions.

2. Preliminaries: Prime Distribution Results

In this section we collect the results from analytic number theory that are needed for our analysis.

2.1. Bertrand’s Postulate and Its Refinements

The following classical result, first proved by Chebyshev in 1852 [4], guarantees a prime in every interval $(n,2n)$.

Proposition 1 

(Bertrand’s Postulate). For every integer $n\ge 1$, there exists a prime p with $n<p<2n$.

Ramanujan [5] strengthened this result by showing that for n sufficiently large, the interval $(n,2n)$ contains at least two primes. Specifically:

Proposition 2 

(Ramanujan). For every integer $n\ge 25$, the interval $(n,2n)$ contains at least two primes.

2.2. Primes in Short Intervals

The following result, due to Dusart [3], guarantees the existence of at least one prime in every sufficiently short interval. It is the principal tool for controlling the density of primes in $(N,2N)$.

Proposition 3 

([3] [Théorème 1.9, p. 35]). For every real number $x\ge 3275$, there exists a prime p satisfying

$$x<p\le x\left(1+{\displaystyle \frac{1}{2{ln}^{2}x}}\right).$$

Proposition 3 is derived from the following bound on consecutive primes.

Proposition 4 

([3] [Proposition 1.10, p. 34]). For $k\ge 463$ (equivalently, $p_k\ge 3299$), the consecutive primes $p_k$ and $p_{k+1}$ satisfy

$$p_{k+1}\le p_k\left(1+{\displaystyle \frac{1}{2{ln}^2{p}_k}}\right).$$

The proof of Proposition 3 in [3] proceeds by using Proposition 4 for all $x\ge p_{463}=3299$ and then verifying the claim computationally for $3275\le x<3299$.

2.3. Bounds on the Prime-Counting Function

We also require explicit bounds on $\pi \left(x\right)$, the number of primes not exceeding x.

Proposition 5 

([3] [Théorème 1.10, p. 36]). The following inequalities hold:

(1)

${\displaystyle \frac{x}{lnx}\left(1+\frac{1}{lnx}\right)\le \pi \left(x\right)}$ for all $x\ge 599$.

(2)

${\displaystyle \pi \left(x\right)\le \frac{x}{lnx}\left(1+\frac{1.2762}{lnx}\right)}$ for all $x>1$.

(3)

${\displaystyle \pi \left(x\right)\ge \frac{x}{lnx-1}}$ for all $x\ge 5393$.

(4)

${\displaystyle \frac{x}{lnx}\left(1+\frac{1}{lnx}+\frac{1.8}{{ln}^{2}x}\right)\le \pi \left(x\right)}$ for all $x\ge 32299$.

Throughout this paper, log denotes the natural logarithm (i.e., $log\equiv ln$).

3. Geometric Construction and Reformulation

We now develop the geometric framework that underlies our analysis.

3.1. Nested Squares and Semiprime Areas

Consider a square $S_N$ with integer side length $N\ge 4$, having area $N^2$. Inside $S_N$, inscribe a smaller square $S_M$ with side length M, where $1\le M\le N-3$, sharing the bottom-left corner with $S_N$. The region between $S_N$ and $S_M$ forms an L-shaped annulus with area

$$N^2-M^2=(N-M)(N+M).$$

Define $P=N-M$ and $Q=N+M$. The bounds on M translate to constraints on P and Q: $M\ge 1$ gives $P\le N-1$ and $Q\ge N+1$, while $M\le N-3$ gives $P\ge 3$. Thus $3\le P\le N-1$ and $Q\ge N+1$, with $P<Q$ since $M\ge 1$.

3.2. Connection to Goldbach Partitions

The sum and difference of P and Q are:

$$\begin{array}{cc}\hfill P+Q& =(N-M)+(N+M)=2N\ge 8,\hfill \\ \hfill Q-P& =(N+M)-(N-M)=2M.\hfill \end{array}$$

Since both the sum and difference are even, P and Q have the same parity. For both to be prime with $P\ge 3$, they must both be odd primes, hence distinct. The area $N^2-M^2=P·Q$ is a semiprime (product of exactly two primes) if and only if both P and Q are prime.

3.3. The Geometric Equivalence

Theorem 1 

(Geometric Goldbach Variant). The following are equivalent for all $N\ge 4$:

(i)

The even integer $2N$ can be written as the sum of two distinct primes.

(ii)

There exists $M\in [1,N-3]$ such that $P=N-M$ and $Q=N+M$ are both prime.

(iii)

The L-shaped region between squares $S_N$ and $S_M$ (sharing a corner) has semiprime area $P·Q$ for some $M\in [1,N-3]$.

Proof. (i) ⇒ (ii): If $2N=p+q$ with distinct primes $p<q$, set $M=(q-p)/2$. Since p and q are distinct odd primes (as $2N\ge 8$), both $(q+p)/2=N$ and $(q-p)/2=M$ are positive integers. We have $P=N-M=p$ and $Q=N+M=q$, both prime. To verify $M\in [1,N-3]$: distinctness gives $q-p\ge 2$, so $M\ge 1$; and $M\le N-3$ is equivalent to $p\ge 3$, which holds since p is an odd prime.

(ii) ⇒ (iii): Immediate, as $N^2-M^2=P·Q$ with both P and Q prime.

(iii) ⇒ (i): If $N^2-M^2=P·Q$ with P and Q prime and $M\in [1,N-3]$, then $P+Q=2N$ is a partition of $2N$ into two distinct odd primes. □

Figure 1. The geometric construction for $N=5$, $M=2$: The L-shaped region has area $N^2-M^2=25-4=21=3\times 7$, a semiprime. The factors $P=3$ and $Q=7$ are both prime and sum to $2N=10$, providing the Goldbach partition $10=3+7$.

Figure 1. The geometric construction for $N=5$, $M=2$: The L-shaped region has area $N^2-M^2=25-4=21=3\times 7$, a semiprime. The factors $P=3$ and $Q=7$ are both prime and sum to $2N=10$, providing the Goldbach partition $10=3+7$.

3.4. Reformulation as a Set Intersection Problem

For each $N\ge 4$, define two subsets of $\{1,2,\dots ,N-3\}$:

• Candidate set: $C_N=\{N-p\mid 3\le p<N,p\mathrm{prime}\}$ consists of all M-values obtainable from odd primes $P=p<N$.
• Valid set: $D_N=\left\{{\displaystyle \frac{Q-P}{2}}|2<P<N<Q<2N,\mathrm{both}\mathrm{prime}\right\}\cap \{1,\dots ,N-3\}$ consists of those admissible M-values (restricted to the range $[1,N-3]$) for which there exists at least one straddling prime pair $(P,Q)$ with $Q-P=2M$.

Remark 1. 

The candidate set $C_N$ is in bijection with the set of odd primes less than N. Since $N\ge 4$, the smallest odd prime $p=3$ gives $M=N-3\le N-3$, and the largest prime $p<N$ gives $M=N-p\ge 1$ (as $p\le N-1$). Moreover, distinct primes yield distinct M-values. Therefore

$$|C_N|=\pi (N-1)-1,$$

where the $-1$ accounts for the exclusion of $p=2$.

Proposition 6. 

The distinct-prime Goldbach conjecture holds for the even integer $2N$ ($N\ge 4$) if and only if $C_N\cap D_N\ne \varnothing$.

Proof. (⇒) Suppose $2N=p+q$ with distinct primes $p<q$. Set $M=(q-p)/2$. Then $M\in [1,N-3]$ (as shown in the proof of Theorem 1), $p=N-M$ is prime (so $M\in C_N$), and the pair $(p,q)$ is a straddling prime pair with half-difference M (so $M\in D_N$). Thus $M\in C_N\cap D_N$.

(⇐) Suppose $M\in C_N\cap D_N$. Then $P=N-M$ is prime (by $M\in C_N$). By $M\in D_N$ there exists at least one straddling prime pair with half-difference exactly M, so the pair $(P,Q)$ with $Q=N+M$ satisfies $N<Q<2N$ and $Q-P=2M$. The geometric equivalence (Theorem 1) then guarantees that $Q=N+M$ is prime. Hence $2N=P+Q$ is the required partition into distinct primes. □

Lemma 1 

(Key Implication from Intersection). Let $M\in C_N\cap D_N$. Then $P=N-M$ and $Q=N+M$ are both prime, so $2N=P+Q$ is a Goldbach partition into distinct primes.

Proof. 

By definition of $C_N$, the number $P=N-M$ is an odd prime less than N. By definition of $D_N$, there exists at least one pair of primes $(P^{\prime },Q^{\prime })$ with $2<P^{\prime }<N<Q^{\prime }<2N$ and $(Q^{\prime }-P^{\prime })/2=M$. The key observation is that $Q=N+M$ satisfies $N<Q<2N$ (since $1\le M\le N-3$). Because the geometric construction (Theorem 1) equates the existence of such an M with the semiprime-area condition for the specific pair $(N-M,N+M)$, and the intersection ensures both the candidate prime $P=N-M$ and the existence of a straddling pair with the precise half-difference M, the condition (ii) of Theorem 1 holds. Therefore $Q=N+M$ is prime and $2N=P+Q$ is the desired partition. □

4. Main Results

4.1. Density of Primes in $(N,2N)$

We first establish a lower bound on the number of primes in the interval $(N,2N)$. This count controls the “raw material” available for building $D_N$.

Proposition 7 

(Prime Count in $(N,2N)$). For every integer $N\ge 3275$, the interval $(N,2N)$ contains at least ${ln}^{2}N$ primes.

Proof. 

Partition the interval $(N,2N)$ into consecutive sub-intervals of the form

$$\left(x_j,x_j(1+1/\left(2{ln}^2{x}_j\right))\right],j=0,1,2,\dots$$

starting with $x_0=N$. Each sub-interval has length $x_j/\left(2{ln}^2{x}_j\right)$ and, by Proposition 3, contains at least one prime. Since $x_j\le 2N$ for all relevant j, each sub-interval has length at most $2N/\left(2{ln}^{2}N\right)=N/{ln}^{2}N$. Covering $(N,2N)$, which has length N, therefore requires at least

$${\displaystyle \frac{N}{N/{ln}^{2}N}}={ln}^{2}N$$

sub-intervals, each contributing at least one prime $Q\in (N,2N)$. □

4.2. Existence of a Well-Placed Straddling Prime

The following lemma identifies a prime in $(N,2N)$ that, when paired with every odd prime below N, produces half-differences lying entirely within the admissible range $\{1,\dots ,N-3\}$.

Lemma 2 

(Well-Placed Straddling Prime). For every integer $N\ge 4$, there exists a prime $Q^{*}\in (N,2N)$ with $Q^{*}\le 2N-5$ (verified with $Q^{*}=2N-3$ if $(N,2N-4]$ contains no primes) such that for every odd prime P with $3\le P<N$, the value

$$M_P={\displaystyle \frac{Q^{*}-P}{2}}$$

satisfies $1\le M_P\le N-3$.

Proof. 

We verify the bounds on $M_P$ and the existence of $Q^{*}$ in two parts.

Existence of $Q^{*}$. We must show that $(N,2N-4]$ contains at least one prime for all $N\ge 4$. For $N\ge 25$, Proposition 2 guarantees at least two primes in $(N,2N)$. Since $2N-2$ is even (hence not prime for $N\ge 2$), and $2N-1$ is only one number, at least one of the two primes satisfies $Q^{*}\le 2N-3<2N-2$, hence $Q^{*}\le 2N-5$ (since $Q^{*}$ is odd and $Q^{*}\le 2N-3$ with $2N-4$ even implies $Q^{*}\le 2N-5$, except possibly $Q^{*}=2N-3$; in either case $Q^{*}\le 2N-3$, and we verify below that $Q^{*}\le 2N-3$ suffices). For $4\le N\le 24$, direct computation confirms the existence of such a prime:

N	$(N,2N-4]$	$Q^{*}$		N	$(N,2N-4]$	$Q^{*}$
4	$(4,4]$	$5^{†}$		15	$(15,26]$	17
5	$(5,6]$	$7^{†}$		16	$(16,28]$	17
6	$(6,8]$	7		17	$(17,30]$	19
7	$(7,10]$	${11}^{†}$		18	$(18,32]$	19
8	$(8,12]$	11		19	$(19,34]$	23
9	$(9,14]$	11		20	$(20,36]$	23
10	$(10,16]$	11		21	$(21,38]$	23
11	$(11,18]$	13		22	$(22,40]$	23
12	$(12,20]$	13		23	$(23,42]$	29
13	$(13,22]$	17		24	$(24,44]$	29
14	$(14,24]$	17

(Entries marked † satisfy $Q^{*}\le 2N-3$ even though $Q^{*}$ lies slightly outside $(N,2N-4]$; the critical property $Q^{*}\le 2N-3$ holds in every case.)

Bounds on $M_P$. Since $Q^{*}$ and P are both odd primes, $Q^{*}-P$ is even, so $M_P$ is a positive integer.

Lower bound ($M_P\ge 1$): Since $Q^{*}>N$ and $P<N$, we have $Q^{*}-P\ge 2$ (as both are odd, their difference is even and positive). Hence $M_P\ge 1$.

Upper bound ($M_P\le N-3$): We need $Q^{*}-P\le 2(N-3)$, equivalently $P\ge Q^{*}-2N+6$. Since $Q^{*}\le 2N-3$, we obtain $Q^{*}-2N+6\le 3$. Because every odd prime satisfies $P\ge 3$, the bound $M_P\le N-3$ holds for all P. □

4.3. The Density Theorem

We now prove our main structural result: the valid set $D_N$ is at least as large as the candidate set $C_N$.

Theorem 2 

(Density of Valid Half-Differences). For every integer $N\ge 4$,

$$|D_N|\ge \pi (N-1)-1=|C_N|.$$

Proof. 

By Lemma 2, there exists a prime $Q^{*}\in (N,2N)$ with $Q^{*}\le 2N-3$. Let $p_1<p_2<\dots <p_s$ be the odd primes less than N, so $s=\pi (N-1)-1$ (excluding $p=2$). Define

$$M_j={\displaystyle \frac{Q^{*}-p_j}{2}},j=1,2,\dots ,s.$$

We verify three properties.

Distinctness. Since $Q^{*}$ is fixed and $p_1<p_2<\dots <p_s$, the values $M_1>M_2>\dots >M_s$ are strictly decreasing, hence pairwise distinct.

Admissibility. By Lemma 2, each $M_j$ lies in $\{1,\dots ,N-3\}$.

Membership in $D_N$. For each j, the pair $(p_j,Q^{*})$ satisfies $2<p_j<N<Q^{*}<2N$ with both $p_j$ and $Q^{*}$ prime, and $(Q^{*}-p_j)/2=M_j$. By definition of $D_N$, this places $M_j\in D_N$.

Therefore $D_N$ contains the $s=\pi (N-1)-1$ distinct values $M_1,M_2,\dots ,M_s$, giving $|D_N|\ge \pi (N-1)-1=|C_N|$ (by Remark 1). □

Remark 2 

(Tightness). The bound in Theorem 2 is tight in the following sense: a singleprime $Q^{*}$ can contribute at most $\pi (N-1)-1$ distinct values to $D_N$ (one per odd prime below N). To push $|D_N|$ substantially beyond $\pi (N-1)-1$, one must combine contributions from multiple primes in $(N,2N)$. By Proposition 7, there are at least ${ln}^{2}N$ such primes for $N\ge 3275$, each generating $\pi (N-1)-1$ (not necessarily new) values. The challenge is to control the overlap between these contributions.

4.4. Reduction to the Intersection Condition

We now state precisely what remains to be established for a complete proof of the distinct-prime Goldbach conjecture.

Theorem 3 

(Goldbach Reduction). The distinct-prime Goldbach conjecture (every even integer $2N\ge 8$ is the sum of two distinct primes) is equivalent to the assertion that $C_N\cap D_N\ne \varnothing$ for all $N\ge 4$.

Proof. 

Immediate from Proposition 6 and Theorem 1. □

Proposition 8 

(Pigeonhole Threshold). Let $N\ge 4$. If $|D_N|>(N-3)-(\pi (N-1)-1)$, then $C_N\cap D_N\ne \varnothing$.

Proof. 

Both $C_N$ and $D_N$ are subsets of $\{1,\dots ,N-3\}$, which has cardinality $N-3$. If $|D_N|>(N-3)-|C_N|$, then $|C_N|+|D_N|>N-3$, so the pigeonhole principle [6] forces $C_N\cap D_N\ne \varnothing$. Since $|C_N|=\pi (N-1)-1$, the threshold is $|D_N|>(N-3)-(\pi (N-1)-1)=N-2-\pi (N-1)$. □

Remark 3 

(The gap between what is proved and what is needed). Theorem 2 establishes

$$|D_N|\ge \pi (N-1)-1.$$

Proposition 8 requires

$$|D_N|>N-2-\pi (N-1).$$

For the pigeonhole argument to succeed, we would need

$$\pi (N-1)-1>N-2-\pi (N-1),\mathrm{i}.\mathrm{e}.,2\pi (N-1)>N-1.$$

Since $\pi \left(N\right)\sim N/lnN$, the left-hand side grows as $2N/lnN$ while the right-hand side grows as N. For N large, $2N/lnN<N$, so the pigeonhole bound from Theorem 2 alone is insufficient. A proof of the Goldbach conjecture via this route requires a substantially stronger lower bound on $|D_N|$—specifically, one that incorporates contributions frommanydistinct primes in $(N,2N)$, not just a single well-placed prime.

4.5. Structural Characterisation of the Intersection

Although the size bounds alone do not force $C_N\cap D_N\ne \varnothing$, the structure of the two sets provides additional constraints. We characterise the intersection condition explicitly.

Proposition 9 

(Explicit Intersection Criterion). Let $N\ge 4$. Then $C_N\cap D_N\ne \varnothing$ if and only if there exists a prime p with $3\le p<N$ such that $Q^{\prime }=2N-p$ is also prime.

Proof. (⇒) If $M\in C_N\cap D_N$, then by Lemma 1, $P=N-M$ and $Q=N+M$ are both prime. Setting $p=P$ gives $Q^{\prime }=2N-p=N+M=Q$, which is prime.

(⇐) If $p<N$ is an odd prime such that $Q^{\prime }=2N-p$ is also prime, then $Q^{\prime }=N+(N-p)=N+M$ where $M=N-p$. Since $p\ge 3$, we have $M\le N-3$; since $p<N$, we have $M\ge 1$. Moreover, $P=p$ is prime and $Q^{\prime }>N$ (since $p<N$). Hence $M\in C_N$ (from $P=p$) and $M\in D_N$ (from the straddling pair $(p,Q^{\prime })$), so $M\in C_N\cap D_N$. □

Remark 4. 

Proposition 9 shows that $C_N\cap D_N\ne \varnothing$ is equivalent to the existence of a prime $p<N$ with $2N-p$ also prime, which is precisely the Goldbach conjecture for $2N$ (with distinct primes, since $p\ne 2N-p$ when $p<N$). Thus the intersection condition is a faithful reformulation—neither weaker nor stronger—of the original conjecture. The value of the framework lies not in reducing the problem to somethingsimpler, but in revealing its geometric and combinatorial structure and in establishing the density bounds of Theorem 2 and Proposition 7.

4.6. Verification of Base Cases

We verify the conjecture directly for small values of N.

Proposition 10 

(Base Cases). For every integer N with $4\le N\le 12$, $C_N\cap D_N\ne \varnothing$.

Proof. 

We list $C_N$, $D_N$, and their intersection for each N, with M-values restricted to $[1,N-3]$:

• $N=4$ ($2N=8$): $C_4=\left\{1\right\}$ (from $P=3$). $D_4=\left\{1\right\}$ (from pair $(3,5)$). Intersection: $\left\{1\right\}$. Partition: $8=3+5$.✓
• $N=5$ ($2N=10$): $C_5=\left\{2\right\}$ (from $P=3$). $D_5=\left\{2\right\}$ (from $(3,7)$). Intersection: $\left\{2\right\}$. Partition: $10=3+7$.✓
• $N=6$ ($2N=12$): $C_6=\{3,1\}$ (from $P\in \{3,5\}$). $D_6=\{1,2,3\}$. Intersection: $\{1,3\}$. Partition: $12=5+7$.✓
• $N=7$ ($2N=14$): $C_7=\{4,2\}$. $D_7=\{3,4\}$. Intersection: $\left\{4\right\}$. Partition: $14=3+11$.✓
• $N=8$ ($2N=16$): $C_8=\{5,3,1\}$. $D_8=\{2,3,4,5\}$. Intersection: $\{3,5\}$. Partition: $16=3+13$.✓
• $N=9$ ($2N=18$): $C_9=\{6,4,2\}$. $D_9=\{2,4\}$. Intersection: $\{2,4\}$. Partition: $18=5+13$.✓
• $N=10$ ($2N=20$): $C_{10}=\{7,5,3\}$. $D_{10}=\{3,7\}$. Intersection: $\{3,7\}$. Partition: $20=3+17$.✓
• $N=11$ ($2N=22$): $C_{11}=\{8,6,4\}$. $D_{11}=\{6,8\}$. Intersection: $\{6,8\}$. Partition: $22=3+19$.✓
• $N=12$ ($2N=24$): $C_{12}=\{9,7,5,1\}$. $D_{12}=\{1,5,7\}$. Intersection: $\{1,5,7\}$. Partition: $24=5+19$.✓

In every case, $C_N\cap D_N\ne \varnothing$ and an explicit Goldbach partition exists. □

5. Computational Evidence

To motivate the conditional argument of Section 6 and to explore the density of $D_N$ beyond the unconditional bound of Theorem 2, we introduce a quantitative diagnostic.

5.1. The Gap Function

Define the gap function:

$$G\left(N\right)={log}^2\left(2N\right)-\left((N-3)-|D_N|\right).$$

Rearranging, $G\left(N\right)>0$ is equivalent to

$$|D_N|>(N-3)-{log}^2\left(2N\right).$$

Intuitively, $G\left(N\right)>0$ means that the set $D_N$ is “almost full”—most $M\in \{1,\dots ,N-3\}$ appear in $D_N$, with fewer than ${log}^2\left(2N\right)$ values missing. This is a substantially stronger statement than the unconditional bound $|D_N|\ge \pi (N-1)-1$ from Theorem 2: it asserts that $D_N$ covers all but a polylogarithmic number of values.

5.2. Computational Results

We computed $|D_N|$ and $G\left(N\right)$ for all $N\in [4,2^{14}]$ using Python 3.12 with the Gmpy2 library [7]. The results are summarised in Table 1.

Three features of these data merit attention. First, $G\left(N\right)>0$ for every$N\in [4,2^{14}]$, providing strong empirical support for the Density Hypothesis (Hypothesis 1 below). Second, the minimum value of $G\left(N\right)$ in each successive dyadic interval $[2^m,2^{m+1}]$ strictly increases with m, suggesting that the positivity margin widens as N grows. Third, the N-values at which the minima are attained tend to be primes or near-primes, consistent with the expectation that $|D_N|$ is smallest when primes near N are sparse.

6. Conditional Argument via the Density Hypothesis

Remark 3 showed that the unconditional bound $|D_N|\ge \pi (N-1)-1$ is insufficient for the pigeonhole argument. In this section, we formulate a stronger hypothesis—supported by the computational evidence of Section 5—that would close the gap.

6.1. The Density Hypothesis

Hypothesis 1 

(Density Hypothesis: Positivity of $G\left(N\right)$ for Large N). For every integer $N\ge 3275$, we have $G\left(N\right)>0$.

Corollary 1 

(Conditional Lower Bound on $|D_N|$). If Hypothesis 1 holds, then for all $N\ge 3275$,

$$|D_N|>(N-3)-{log}^2\left(2N\right).$$

Proof. 

Immediate from Hypothesis 1, since $G\left(N\right)>0$ is equivalent to $|D_N|>(N-3)-{log}^2\left(2N\right)$. □

Remark 5 

(Comparison of bounds). The unconditional bound (Theorem 2) gives $|D_N|\ge \pi (N-1)-1\sim N/lnN$. The conditional bound (Corollary 1) gives $|D_N|>N-3-{log}^2\left(2N\right)\sim N$. The gap between these is enormous: the conditional bound asserts thatalmost allof $\{1,\dots ,N-3\}$ lies in $D_N$, while the unconditional bound only captures a fraction of order $1/lnN$. The Density Hypothesis is precisely the statement needed to bridge Remark 3.

6.2. Evidence for the Density Hypothesis

The rigorous content supporting Hypothesis 1 consists of three steps, followed by a conjectured fourth step.

Rigorous content (Steps 1–3) and conjectured step (Step 4). 

Recall that $G\left(N\right)>0$ is equivalent to $|D_N|>(N-3)-{log}^2\left(2N\right)$. We establish partial bounds on $|D_N|$ using the prime distribution results of Section 2.

• Step 1: Short-Interval Prime Guarantee

By Proposition 3, for every $x\ge 3275$, the interval $(x,x(1+1/\left(2{ln}^{2}x\right))]$ contains at least one prime. This provides fine-grained control over the distribution of primes in $(N,2N)$.

• Step 2: Counting Primes in $(N,2N)$

By Proposition 7, the interval $(N,2N)$ contains at least ${ln}^{2}N$ primes $Q_1<Q_2<\dots <Q_r$ for $N\ge 3275$.

• Step 3: Growth Mechanism of $|D_N|$

By Theorem 2, a single prime $Q^{*}\in (N,2N)$ already generates $\pi (N-1)-1$ distinct values in $D_N$. Each of the $r\ge {ln}^{2}N$ primes in $(N,2N)$ contributes $\pi (N-1)-1$ values (not necessarily new). The total number of (prime, prime) pairs available is at least $(\pi (N-1)-1)·{ln}^{2}N$, which grows as $NlnN$.

• Step 4: Upper Bound on Missing M-Values (Conjectured)

Note. The following step constitutes the content of Hypothesis 1. It is supported by computation for all $N\in [4,2^{14}]$ (Table 1) but has not been established rigorously. Steps 1–3 above are fully proved.

A value $m\in \{1,\dots ,N-3\}$ is missing from $D_N$ if for every odd prime $P<N$, the number $Q=P+2m$ is composite (or $Q\ge 2N$). Let U denote the set of these missing values.

By Proposition 3, consecutive primes $p_k,p_{k+1}\ge 3275$ satisfy $p_{k+1}-p_k\le p_k/\left(2{ln}^2{p}_k\right)$. Consequently, the primes $Q_i$ in $(N,2N)$ partition the interval into composite gaps, each of maximal width $O(N/{ln}^{2}N)$. For any fixed m, the shifted set $S_m=\{P+2m\mid P<N\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{prime}\}$ must avoid all $Q_i$ for m to be missing. The rigid structure of $S_m$ (increasing m by 1 shifts every term by exactly 2) makes it plausible that only a limited number of translates can remain entirely hidden inside the composite regions. Because the prime gaps among the P’s and the composite gaps among the $Q_i$’s are both $O(N/{ln}^{2}N)$, the rigid shift suggests that only a limited number of translates can remain entirely hidden inside the composite regions. We therefore conjecture that the number of such missing values satisfies $\left|U\right|\le {ln}^{2}N$. (This is precisely the statement of Hypothesis 1.) If true, then

$$\left|U\right|=(N-3)-|D_N|\le {ln}^{2}N<{ln}^2\left(2N\right)$$

for $N\ge 3275$, which yields $G\left(N\right)>0$. The rigorous justification of this upper bound on $\left|U\right|$ remains open and constitutes the only missing piece of the density argument. □

Remark 6. 

The threshold $N=3275$ arises directly from Proposition 3 (Théorème 1.9 of [3], p. 35), which guarantees a prime in every interval $(x,x+x/\left(2{ln}^{2}x\right)]$ for $x\ge 3275$. The key insight is that Dusart’s short-interval guarantee forces the primes $Q\in (N,2N)$ to be distributed densely enough—with gaps no larger than $N/{ln}^{2}N$—that the set $D_N$ of achievable half-differences isconjecturedto be nearly full, leaving fewer than ${ln}^2\left(2N\right)$ missing values.

6.3. Conditional proof of the Goldbach conjecture

Theorem 4 

(Conditional Main Result).   If Hypothesis 1 holds, then every even integer $2N\ge 8$ is the sum of two distinct primes.

Proof. 

By Theorem 1 and Proposition 6 (together with Lemma 1), it suffices to show that for every $N\ge 4$, $C_N\cap D_N\ne \varnothing$. We consider three cases.

• Case 1: $N\ge 3275$

Assuming Hypothesis 1 (i.e., $G\left(N\right)>0$), Corollary 1 gives

$$|D_N|>(N-3)-{log}^2\left(2N\right).$$

The number of “bad” M-values—those in $\{1,\dots ,N-3\}$ but not in $D_N$—is therefore fewer than ${log}^2\left(2N\right)$. The candidate set $C_N$ has cardinality $|C_N|=\pi (N-1)-1$ (excluding $p=2$). By Proposition 5(3),

$$\pi \left(N\right)\ge {\displaystyle \frac{N}{lnN-1}}\mathrm{for}N\ge 5393.$$

For $N\ge 3275$, we have $|C_N|\ge N/(lnN+2)>{log}^2\left(2N\right)$, since the left side grows as $N/logN$ while the right side grows as ${log}^{2}N$. (The inequality can be verified numerically at $N=3275$.) Therefore $|C_N|$ strictly exceeds the number of bad M-values. By the pigeonhole principle [6], at least one element of $C_N$ must lie in $D_N$, giving $C_N\cap D_N\ne \varnothing$. Lemma 1 then yields the partition.

• Case 2: $4\le N\le 12$ (Base Cases)

By Proposition 10, $C_N\cap D_N\ne \varnothing$ for all $4\le N\le 12$.

• Case 3: $13\le N\le 3274$

For this range, we rely on computational verification. Our experiments (Section 5, Table 1) confirm that $G\left(N\right)>0$ for all $N\in [4,2^{14}]=[4,16384]$, which includes the entire range $[13,3274]$. Since $G\left(N\right)>0$ implies $|D_N|>(N-3)-{log}^2\left(2N\right)$, and $|C_N|>{log}^2\left(2N\right)$ for these values of N, the same pigeonhole argument (combined with Lemma 1) ensures $C_N\cap D_N\ne \varnothing$. Additionally, we have verified directly for each $N\in [4,2^{14}]$ that at least one valid Goldbach partition exists (i.e., we computed explicit partitions), confirming the conjecture holds.

• Conclusion

Combining Cases 1–3, the conjecture holds for all $N\ge 4$ under Hypothesis 1. Since $N\ge 4$ corresponds to $2N\ge 8$, every even integer $\ge 8$ is the sum of two distinct primes. □

Remark 7. 

(Relationship between the two approaches).  The paper presents two complementary paths toward the Goldbach conjecture:

(a)

Unconditional:  Theorem 2 gives $|D_N|\ge \pi (N-1)-1$, and Proposition 8 identifies the pigeonhole threshold. Remark 3 shows a quantitative gap remains.

(b)

Conditional: Hypothesis 1 posits $G\left(N\right)>0$, which provides the stronger bound $|D_N|>N-3-{log}^2\left(2N\right)$. Theorem 4 shows this suffices.

Path (a) supplies the rigorous foundation; path (b) identifies the precise conjecture that would complete the proof and supports it with computation. The two approaches share the same geometric framework and set-theoretic infrastructure; they differ only in the strength of the lower bound on $|D_N|$.