Goldbach Conjecture Verifier
Every even integer ≥ 4 is the sum of two primes — verify it yourself
Max 10,000 for performance
The Goldbach Comet plots the number of prime pairs (p, q) with p + q = 2n for each even number 2n. The comet-like structure appears naturally — more pairs for larger numbers.
Each dot = one even number. Y-axis = number of Goldbach prime pairs. The linear growth hints at the conjecture's deep truth.
About Goldbach's Conjecture
In 1742, Christian Goldbach wrote a letter to Leonhard Euler proposing that every integer greater than 2 can be expressed as the sum of three primes. Euler reformulated it to what we now call the Goldbach Conjecture: every even integer greater than 2 is the sum of two primes.
Examples:
The conjecture has been verified for all even numbers up to 4 × 10¹⁸ (4 quintillion) by computer. Despite this, it remains unproven — one of the oldest unsolved problems in mathematics, with a $1 million Millennium Prize offered for a proof.
Frequently Asked Questions
Has Goldbach's Conjecture been proven?
No. Despite being verified for all even numbers up to 4 × 10¹⁸ and being one of the most famous conjectures in mathematics, no general proof or disproof exists. It remains one of the great open problems in number theory.
What is the "weak" Goldbach Conjecture?
The weak (or ternary) Goldbach Conjecture states that every odd number ≥ 7 is the sum of three odd primes. This was proved by Harald Helfgott in 2013, while the strong (binary) conjecture remains open.
Why is 1 not considered a prime in this context?
By modern convention, 1 is not a prime number (the fundamental theorem of arithmetic requires unique factorization). Under Goldbach's original formulation with 1 as prime, the statement changes slightly — modern mathematics uses the definition excluding 1.
What is the Goldbach Comet?
The Goldbach Comet is a plot of the number of ways each even number can be written as the sum of two primes (called Goldbach pairs). When plotted, the data forms a comet-like shape that branches and grows roughly linearly — a beautiful visual representation of the conjecture's structure.
Is there a prize for proving Goldbach's Conjecture?
A $1 million prize was offered by publisher Faber & Faber in 2000 to promote the novel "Uncle Petros and Goldbach's Conjecture." That prize has expired, but a proof would certainly earn the solver recognition as one of the greatest mathematicians in history.
What is the closest result to a proof?
Vinogradov (1937) proved every sufficiently large odd number is the sum of three primes. Chen Jingrun (1966) proved every sufficiently large even number is the sum of a prime and a semiprime (product of at most two primes). These are the closest known results.
Does the number of Goldbach pairs always increase?
Not monotonically — the count fluctuates for individual numbers. But the average number of Goldbach representations grows roughly linearly with the number, as predicted by the Hardy-Littlewood conjectures on the distribution of primes.
Why is 2 a special case?
The conjecture applies to even numbers ≥ 4. The only even prime is 2, so 4 = 2+2 is the only case that uses the even prime. All larger even Goldbach representations involve two odd primes.