Goldbach Conjecture Calculator
Enter an even integer greater than 2 to test Goldbach’s conjecture and find prime pair decompositions.
Performance limit: maximum input is 2,000,000 for fast in-browser calculations.
The Goldbach conjecture is one of the oldest unsolved problems in number theory. Despite centuries of effort, no counterexample has ever been found: every tested even integer greater than 2 can be written as the sum of two prime numbers. This calculator lets you verify that claim for specific values and explore the prime pairs behind each result.
What Is Goldbach’s Conjecture?
The version most people mean is the strong Goldbach conjecture: for every even integer n > 2, there exist primes p and q such that: n = p + q.
Example:
- 10 = 3 + 7
- 28 = 5 + 23
- 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53
How to Use This Calculator
- Type any even integer greater than 2.
- Choose whether to show just one valid pair or all unique prime pairs.
- Click Find Prime Pair(s) to run the check.
- Optionally click Use Random Even n for quick exploration.
How the Algorithm Works
1) Prime Generation
The calculator uses the Sieve of Eratosthenes to mark all prime numbers up to your input value. This is much faster than testing primality repeatedly for each candidate.
2) Pair Search
It scans values from 2 to n/2 and checks whether both p and n - p are prime. This naturally avoids duplicate orderings (for example, 3 + 97 and 97 + 3 are counted once).
3) Output
If at least one pair exists, the calculator confirms the conjecture for that specific number and optionally lists every unique representation.
Why This Problem Still Matters
Goldbach’s conjecture sits at the intersection of simple statements and deep mathematics. Even though computers have verified it for enormous ranges, a complete proof for all even integers has not been found. This is a great example of the difference between:
- Empirical verification (tested cases), and
- Formal proof (true for infinitely many cases).
Frequently Asked Questions
Is this a proof of Goldbach’s conjecture?
No. It verifies individual values (or many values if you test repeatedly), but a full proof must cover every valid integer without exception.
Why only even numbers greater than 2?
That is the exact domain of the strong Goldbach conjecture.
What counts as a unique pair?
Pairs are unordered in this calculator. So (11, 89) is the same as (89, 11) and is listed once.
Can I use very large numbers?
This page runs in your browser, so practical speed and memory matter. The limit here is set to 2,000,000 to keep responses fast on typical devices.
Try It Yourself
Start with values like 20, 100, 1000, or 10000 and inspect how many prime decompositions appear. You’ll quickly notice that many even numbers have multiple Goldbach pairs, which makes this conjecture both intuitive and surprisingly resilient.