Experimental Results on Goldbach's Conjecture

This work builds upon Geometric Insights into the Goldbach Conjecture.

Author: Frank Vega
Institution: Information Physics Institute, Hialeah, FL, USA

Keywords: Goldbach conjecture, geometric construction, semiprimes, pigeonhole principle

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Abstract

The Goldbach conjecture states that every even integer greater than 2 is the sum of two primes. We present a computational approach that provides strong evidence for a variant: every even integer $\geq 8$ is the sum of two distinct primes.

Our key insight is a geometric equivalence: this is true if and only if for every $N \geq 4$, there exists an integer $M$ such that the L-shaped region $N^2 - M^2$ between nested squares has a semiprime area $P \cdot Q$, where $P = N - M$ and $Q = N + M$ are both prime.

Through computational analysis up to $N = 2^{14}$ and application of the pigeonhole principle, we demonstrate this variant holds for all $N \geq 4$ within our verified range and provide strong theoretical evidence for its general validity.

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1. Introduction

The Goldbach conjecture is one of mathematics' oldest unsolved problems: can every even integer greater than 2 be expressed as the sum of two primes?

We study a variant that excludes identical primes:

Variant: Every even integer $\geq 8$ is the sum of two distinct primes.

This excludes $4 = 2 + 2$ and $6 = 3 + 3$ while preserving the essence of the original conjecture.

We provide strong computational and theoretical evidence for this variant by connecting it to a surprising geometric property of nested squares.

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2. The Geometric Connection

Construction

Start with a square $S_N$ of side length $N \geq 4$. Inside it, place a smaller square $S_M$ of side length $M$ (where $1 \leq M \leq N-3$) sharing the same corner. The L-shaped region between them has area:

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

Let $P = N - M$ and $Q = N + M$. Then:

• $P + Q = 2N$ (an even number)
• $P \cdot Q = N^2 - M^2$ (the L-shaped area)
• Both $P$ and $Q$ must be odd (same parity)

The Key Equivalence

The Goldbach variant is true $\Longleftrightarrow$ For every $N \geq 4$, there exists an $M$ making both $P$ and $Q$ prime.

When this happens, the L-shaped area is a semiprime (product of exactly two primes).

Figure 1: The L-shaped region between nested squares. For N=5, M=2: P=3 and Q=7 (both prime), giving area 21 = 3×7 and sum 3+7=10.

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3. Why This Connection Matters

For any even number $2N$, finding a Goldbach partition means finding primes $P$ and $Q$ where $P + Q = 2N$.

Geometrically, this is equivalent to finding an $M$ value such that:

• $P = N - M$ is prime
• $Q = N + M$ is prime
• The L-shaped area $P \cdot Q$ is a semiprime

This transforms an arithmetic problem into a geometric search.

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4. Computational Evidence

Defining the Set $D_N$

For each $N$, define $D_N$ as the set of all valid $M$ values that create prime pairs:

$$D_N = \left\{M = \frac{Q - P}{2} \mid P, Q \text{ are prime, } 2 < P < N < Q < 2N\right\} \cap \{1,\dots,N-3\}$$

Question: How many valid $M$ values exist for each $N$?

Gap Function

We define a "gap function":

$$G(N) = \log^2(2N) - ((N-3) - |D_N|)$$

This measures how many "bad" $M$ values exist (those that don't produce prime pairs) compared to the logarithmic bound.

Experimental Results

We computed $|D_N|$ for all $N$ from 4 to $2^{14}$ (16,384). Key findings:

Table 1: Minimum Gap Values Across Power-of-Two Intervals

Interval	Range	Min at $N$	Min $G(N)$
2	$[4, 8]$	5	4.30
3	$[8, 16]$	9	7.36
4	$[16, 32]$	19	10.24
5	$[32, 64]$	61	14.08
6	$[64, 128]$	73	17.84
7	$[128, 256]$	151	20.61
8	$[256, 512]$	269	23.54
9	$[512, 1024]$	541	28.81
10	$[1024, 2048]$	1327	33.15
11	$[2048, 4096]$	2161	35.08
12	$[4096, 8192]$	7069	42.33
13	$[8192, 16384]$	14138	44.06

Key Observation: $G(N) &gt; 0$ always, and the minimum increases with each interval!

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5. Theoretical Framework and Evidence

Main Result

Claim: Our computational evidence strongly suggests that every even integer $\geq 8$ is the sum of two distinct primes.

Strategy

The computational data shows that $G(N) &gt; 0$, which means:

$$|D_N| > (N-3) - \log^2(2N)$$

In other words, the number of "bad" $M$ values is less than $\log^2(2N)$.

Now, for each prime $P \in [3, N-1]$, we get a candidate $M = N - P$. There are $\pi(N-1) - 1$ such candidates (where $\pi$ counts primes).

Pigeonhole Principle: If we have more candidates than bad values, at least one candidate must be good!

For $N \geq 6$: $\pi(N) &gt; \frac{N}{\log N + 2}$

For $N \geq 328$: $\frac{N}{\log N + 2} &gt; \log^2(2N)$

Therefore: candidates > bad values $\Longrightarrow$ at least one good $M$ exists!

Base Cases

For $N = 4$ to $12$, we verify directly (additional examples included for illustration):

• $N=4$ ($2N=8$): $C_4 = \{1\}$ (from $P=3$). $D_4 = \{1\}$ (from pair $(3,5) $). Intersection: $\{1\}$. Partition: $8 = 3+5$. ✓
• $N=5$ ($2N=10$): $C_5 = \{2\}$ (from $P=3$). $D_5 = \{2\}$ (from $(3,7) $). Intersection: $\{2\}$. Partition: $10 = 3+7$. ✓
• $N=6$ ($2N=12$): $C_6 = \{3,1\}$ (from $P \in \{3,5\}$). $D_6 = \{1,2,3\}$ ($M=4$ excluded as $&gt;3$). Intersection: $\{1,3\}$. Partition: $12 = 5+7$. ✓
• $N=7$ ($2N=14$): $C_7 = \{4,2\}$. $D_7 = \{3,4\}$ ($M=5$ excluded as $&gt;4$). Intersection: $\{4\}$. 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: $24 = 5+19$. ✓

For $13 \leq N \leq 3274$, the conjecture holds by direct computational verification (included in our analysis up to $N=2^{14}$).

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6. Conclusion

We have demonstrated through computational and theoretical analysis that every even integer $\geq 8$ is potentially the sum of two distinct primes by:

1. Establishing a geometric equivalence with nested squares and semiprimes
2. Computing empirical bounds on the number of valid configurations up to $N = 2^{14}$
3. Applying the pigeonhole principle to provide strong theoretical evidence that at least one solution exists for all $N$

This demonstrates how geometric thinking and computational data can combine with classical combinatorial principles to provide compelling evidence for number-theoretic claims.

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Code and Data

The computational verification is available in this repository. Run python experiment.py to reproduce the results in Table 1.

Requirements: Python 3.12+, gmpy2 library

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Documentation

Available as PDF at Geometric Insights into the Goldbach Conjecture.