Goldbach Conjecture Calculator
Express any even number as the sum of two prime numbers.
Enter an even number ≥ 4 (max 10,000,000)
Goldbach's Conjecture (1742) states that every even integer greater than 2 can be written as the sum of two prime numbers. This calculator finds all such prime pairs for any given even number.
Algorithm: For an even number n, we iterate through all primes p from 2 to n/2. For each prime p, we check if (n - p) is also prime. If both conditions are met, we have found a valid Goldbach pair.
Example: For n = 28, the pairs are: 5 + 23, 11 + 17. Both 5 and 23 are prime, and 5 + 23 = 28. Similarly, 11 and 17 are prime, and 11 + 17 = 28.
Verification Status: The conjecture has been verified for all even numbers up to 4 × 1018, but a general proof remains elusive.
What is Goldbach's Conjecture?
Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and 10 = 3 + 7 = 5 + 5. Despite being proposed in 1742, it remains one of the oldest unsolved problems in mathematics.
Has Goldbach's Conjecture been proven?
No, Goldbach's Conjecture has not been proven despite nearly 300 years of effort by mathematicians. However, it has been verified computationally for all even numbers up to 4 × 10^18 (4 quintillion). The closest result is Chen's theorem (1966), which proves every sufficiently large even number is the sum of a prime and a semiprime (product of at most two primes).
Why is Goldbach's Conjecture important?
Goldbach's Conjecture is important because it connects two fundamental concepts in number theory: even numbers and prime numbers. Its simplicity makes it accessible to anyone, yet its proof has eluded the greatest mathematicians. Solving it would likely require new mathematical techniques that could advance our understanding of prime number distribution.
Can an even number have multiple Goldbach representations?
Yes, most even numbers can be expressed as the sum of two primes in multiple ways. For example, 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. The number of representations generally increases as the even number gets larger, though this growth is irregular.