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Week 3 · ClassicalPartial progress
Primary source

Goldbach's Conjecture — Every Even Number as a Sum of Two Primes

Simple rule · open problem· Posed 1742A002375

Problem

Can every even integer n>2n > 2 be written as the sum of two primes? E.g., 10=3+7=5+510 = 3+7 = 5+5, 100100 has 6 representations.

Why it matters

The problem traces to Goldbach's letter to Euler on 1742-06-07. More than 280 years later it remains open. Large even numbers tend to have many representations: 100100 has 6, 1,0001{,}000 has 28, and 1,000,0001{,}000{,}000 has 5,402 unordered representations. The count is not monotone for every even number, and abundant computation is not a universal proof.

Progress so far

Verified up to 4×10184 \times 10^{18} (Oliveira e Silva 2014). Vinogradov (1937): every sufficiently large odd number is a sum of three primes. Harald Helfgott (2013): every odd number >5> 5 is a sum of three primes — ternary Goldbach completely proved. The strong (even) Goldbach remains open.

Further reading

💡 Explore together, one line at a time(0 contributions)

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What did you notice?

You do not need a complete proof. A small observation can open the next path.

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