How does this Goldbach calculator work?

For a given even integer N, the calculator generates all prime numbers up to N and then checks each prime p to see whether N − p is also prime. Each pair (p, N − p) with p ≤ N − p is listed as one Goldbach decomposition.

What is the difference between the strong and weak Goldbach conjectures?

The strong (binary) Goldbach conjecture says every even integer greater than 2 is the sum of two primes. The weak (ternary) Goldbach conjecture says every odd integer greater than 5 is the sum of three primes. The weak conjecture was proven in 2013 by Harald Helfgott; the strong conjecture is still open.

Up to what number has Goldbach’s conjecture been checked?

As of recent published computations, the strong Goldbach conjecture has been verified for all even integers up to at least 4 × 10^18. Our online calculator is limited to more modest ranges for performance reasons, but the underlying conjecture has been checked far beyond that.

Goldbach Conjecture Calculator & Visualizer

What is Goldbach’s conjecture?

Goldbach’s conjecture is one of the oldest and most famous unsolved problems in number theory. In its modern (strong) form it states:

Every even integer N > 2 can be written as the sum of two prime numbers.

For example:

4 = 2 + 2
10 = 3 + 7 = 5 + 5
28 = 5 + 23 = 11 + 17
100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

The calculator above lets you experiment with this statement by listing all prime pairs \( (p, q) \) such that \( p + q = N \) for a chosen even number \( N \).

Strong vs. weak Goldbach conjecture

There are two closely related statements usually called Goldbach’s conjectures:

Strong (binary) Goldbach conjecture

The version implemented by this tool:

For every even integer \( N > 2 \), there exist primes \( p, q \) such that \( N = p + q \).

Weak (ternary) Goldbach conjecture

The weak conjecture says that every odd integer greater than 5 can be written as the sum of three primes:

For every odd integer \( n > 5 \), there exist primes \( p, q, r \) such that \( n = p + q + r \).

The weak conjecture was proven in 2013 by Harald Helfgott, using a combination of analytic number theory and extensive computation. The strong conjecture, however, remains open.

How the calculator finds Goldbach pairs

For a given even integer \( N \), the algorithm used in this page is:

Generate all prime numbers up to \( N \) using a fast sieve (Sieve of Eratosthenes).
For each prime \( p \le N \), compute \( q = N - p \).
If \( q \) is also prime and \( p \le q \), record the pair \( (p, q) \).

This ensures that each unordered pair \( \{p, q\} \) is listed only once when “Show each pair once” is enabled.

Complexity and limits

The sieve runs in roughly \( O(N \log \log N) \) time and uses \( O(N) \) memory, which is very fast for values up to about 1,000,000 in a modern browser. That is why the input is capped at 1,000,000.

What is known about Goldbach’s conjecture?

Although no complete proof is known, mathematicians have made substantial progress:

Computational verification: The conjecture has been checked for all even numbers up to at least \( 4 \times 10^{18} \) (4 quintillion) using large-scale computations.
Almost all even numbers: Analytic results show that “almost all” even integers satisfy Goldbach’s conjecture, in the sense that the set of exceptions has density zero.
Weaker forms: The weak Goldbach conjecture (three primes for odd numbers) is proven.

Despite this, a general proof that every even integer greater than 2 is the sum of two primes is still missing.

Patterns you can explore with the tool

By sliding through even numbers and watching the list of prime pairs, you can observe several patterns:

The number of decompositions tends to grow as \( N \) increases.
Many decompositions involve the small prime 3 or 5.
Pairs are symmetric: if \( N = p + q \), then also \( N = q + p \), but we count each unordered pair once.

FAQ

Is Goldbach’s conjecture proven?

No. It is supported by overwhelming numerical evidence and strong partial results, but a full proof (or counterexample) is still unknown.

Could this calculator find a counterexample?

In principle, if you tried an even number that had no prime decomposition, that would be a counterexample. In practice, our range (up to 1,000,000) is far below the limits already checked by professional computations, so you will not find a new counterexample here.

Why do we only consider even numbers?

The strong Goldbach conjecture is specifically about even integers. Odd integers are covered by the weak (three-prime) version instead.

Can an even number have many different Goldbach decompositions?

Yes. For example, 100 has six distinct decompositions into two primes. As numbers grow, the number of decompositions generally increases, though irregularly.