Goldbach Conjecture Verifier

Welcome to the Goldbach Conjecture Verifier, an interactive tool that confirms one of the oldest open problems in number theory for any even integer greater than 2. Enter your number and instantly see every pair of primes that sums to it, the value of the Goldbach partition function g(n), and the famous Goldbach comet plot. The bridge diagram and comet chart make the structure behind the 1742 conjecture visually intuitive.

What is the Goldbach Conjecture?

The Goldbach conjecture is a statement in number theory proposed by the Prussian mathematician Christian Goldbach in a letter to Leonhard Euler on 7 June 1742. In its modern form it states:

For example: \(4 = 2 + 2\), \(6 = 3 + 3\), \(8 = 3 + 5\), \(10 = 3 + 7 = 5 + 5\), \(100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53\).

Despite its simple statement, the conjecture has remained unproven for almost three centuries. It has been computationally verified for every even integer up to \(4 \times 10^{18}\) as of recent large-scale efforts, but a general proof still escapes mathematicians.

The Goldbach Partition Function g(n)

For an even integer \(n\), the number of distinct unordered pairs of primes that sum to \(n\) is denoted \(g(n)\), the Goldbach partition function:

The Goldbach conjecture is equivalent to the claim that \(g(n) \ge 1\) for every even \(n > 2\). Plotted against \(n\), the values of \(g(n)\) form a visually striking figure known as the Goldbach comet — a dense, bright band of points that fans out as \(n\) grows. Distinct horizontal bands appear within the comet: numbers divisible by 6 tend to sit higher than numbers divisible only by 2, because more small primes are available as summands.

How to Use This Verifier

1. Enter an even integer greater than 2. Click a quick example (100, 1,000, 10,000, 123,456, 1,000,000) or type your own.
2. Click "Verify Goldbach". The tool finds every prime pair that sums to your number using a sieve of Eratosthenes.
3. Read the verdict. The green banner confirms the conjecture holds for your number, and the hero panel reports \(g(n)\).
4. Study the bridge diagram. Each prime pair is drawn as two colored segments on a 0-to-\(n\) line, with the red center marker at \(n/2\). Pairs near the center are more balanced.
5. Explore the comet. The scatter chart shows \(g(m)\) for even \(m\) close to your input, highlighting your number in red so you can see where it sits in the comet pattern.
6. Scan the full pair table. Every \((p, q)\) pair is listed with the difference \(q - p\). Copy all pairs with one click.

What Makes a Pair Special?

• Smallest-p pair — The pair that uses the smallest prime \(p\). Often this is \(3\) or \(5\) for moderate \(n\). When \(n\) is a power of 2 plus 2, it can be \(2 + (n-2)\) itself.
• Most balanced pair — The pair with \(p\) closest to \(n/2\). When both primes equal \(n/2\), \(n\) must be twice a prime (e.g., \(10 = 5 + 5\), \(14 = 7 + 7\), \(26 = 13 + 13\)).
• Largest-p pair — The pair with the largest \(p\) such that \(p \le q\). This is the "most balanced from the other side" and gives a visual bound on how close to \(n/2\) primes cluster.

Goldbach by the Numbers

Classic partition counts

Asymptotic behavior

Heuristic arguments from the Hardy–Littlewood conjecture suggest that \(g(n)\) grows roughly like

where \(C_2 \approx 0.66016\) is the twin prime constant. The extra product reflects why even numbers with many small prime factors (multiples of 6, 30, and so on) tend to have disproportionately many Goldbach pairs — the source of the horizontal bands in the comet.

Weak vs Strong Goldbach

• Strong (binary) Goldbach conjecture — every even \(n > 2\) is a sum of two primes. Still open.
• Weak (ternary) Goldbach conjecture — every odd \(n > 5\) is a sum of three primes. Proved by Harald Helfgott in 2013, completing a decades-long program initiated by Vinogradov in 1937.

The strong form implies the weak form: if every even \(n\) is a sum of two primes, then every odd \(n > 5\) is that sum plus an extra \(3\). The converse, unfortunately, is not known to hold.

Famous Partial Results

• 1923 — Hardy & Littlewood: assuming the Generalized Riemann Hypothesis, almost every even integer is a sum of two primes.
• 1937 — Ivan Vinogradov: proved the ternary conjecture for all sufficiently large odd integers.
• 1973 — Chen Jingrun: every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes (Chen's theorem).
• 1995 — Olivier Ramaré: every even integer is the sum of at most 6 primes.
• 2013 — Harald Helfgott: proved the weak Goldbach conjecture unconditionally.
• 2014 — Oliveira e Silva, Herzog & Pardi: the strong conjecture verified for all even \(n \le 4 \times 10^{18}\).

Frequently Asked Questions

What is the Goldbach conjecture?

The Goldbach conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers. It was first stated by Christian Goldbach in 1742 and has been verified for astronomically large numbers but never proved in general.

Has the Goldbach conjecture been proven?

No. As of 2026 the strong Goldbach conjecture remains an open problem. The weak (ternary) version — every odd integer greater than 5 is the sum of three primes — was proved by Harald Helfgott in 2013.

What is the Goldbach partition function g(n)?

\(g(n)\) is the number of unordered pairs of primes that sum to \(n\). For example \(g(10) = 2\) because \(10 = 3 + 7 = 5 + 5\). The Goldbach conjecture is the statement that \(g(n) \ge 1\) for every even \(n > 2\).

Why does the Goldbach conjecture only apply to even integers?

Every prime except \(2\) is odd. Odd + odd = even, so sums of two odd primes are always even. Odd integers are handled by the ternary Goldbach conjecture, which asks about sums of three primes.

What is the Goldbach comet?

The Goldbach comet is a scatter plot of \(g(n)\) versus \(n\). It has a famous tail-like, banded shape. Horizontal bands appear because even numbers with many small prime divisors tend to have proportionally more partitions.

How many prime pairs sum to 100?

There are six: \(3+97\), \(11+89\), \(17+83\), \(29+71\), \(41+59\), \(47+53\). So \(g(100) = 6\). Try 100 in the verifier above to see each pair visualized.

Additional Resources

• Goldbach's conjecture — Wikipedia
• Goldbach's comet — Wikipedia
• Goldbach Conjecture — MathWorld