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Mind School·Wonder·Honor-system

The Goldbach Conjecture

Every even number greater than two is the sum of two primes. Try it. Now prove it.

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Characterization

On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he conjectured that every integer greater than 2 can be expressed as the sum of three primes. Euler reformulated the conjecture in the margin: every even integer greater than 2 is the sum of two primes. This is the strong Goldbach conjecture, and it has been verified computationally for every even number up to 4 × 10¹⁸ by Tomás Oliveira e Silva (2013). It has never been proved. The conjecture is playable with pencil and paper: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, and so on. Every even number you test will satisfy it. The weak Goldbach conjecture — that every odd integer greater than 5 is the sum of three primes — was proved by Harald Helfgott in 2013, building on a tradition stretching from Hardy and Littlewood's circle method through Vinogradov's 1937 result for sufficiently large odd numbers. But the strong conjecture remains beyond reach. The analytic methods that proved the weak version lose their power in the strong case; new ideas appear to be needed. The Academy hosts the Goldbach conjecture in the Mind School because it is the archetype of a mathematical wonder: a statement a child can test, a pattern that holds without exception to the limits of computation, and a truth — if it is a truth — that no one alive can demonstrate.

Lineage

Christian Goldbach to Leonhard Euler, letter of 7 June 1742. Euler's reformulation in his reply. Ivan Vinogradov, "Representation of an Odd Number as a Sum of Three Primes," Doklady Akademii Nauk SSSR 15, 1937. Harald Helfgott, "The Ternary Goldbach Conjecture Is True," arXiv:1312.7748, 2013. Tomás Oliveira e Silva, "Goldbach Conjecture Verification," computational project, 2013 (verified to 4 × 10¹⁸). G. H. Hardy and J. E. Littlewood, "Some Problems of 'Partitio Numerorum'; III," Acta Mathematica 44, 1923. Yuan Wang, Goldbach Conjecture (World Scientific, 2nd ed., 2002).

Quests

Three quests — one for each archetype. Choose the one that fits your way of taking up the discipline.

  • For every even integer from 4 to 200, compute the number of distinct Goldbach partitions — the number of ways it can be written as a sum of two primes (counting order once). Plot or tabulate the results. Identify any patterns in the growth of the partition count. Compare your observations to the Goldbach comet — the scatter plot of partition counts — and to Hardy and Littlewood's 1923 conjectural asymptotic formula. Record the table, the pattern observed, and a note on what the data suggests about why the conjecture is believed to be true.

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  • Select twenty even numbers — at least five of them above 100 — and find, for each, a decomposition as a sum of two primes. Work by hand or with minimal computational aid. Record each decomposition. Attend to the experience: does finding the pair feel easy, laborious, or uncertain? At what magnitude does the search begin to feel different?

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  • Explain the Goldbach conjecture to a reader who knows what prime numbers are. State the strong and weak conjectures; explain the difference. Cite Goldbach's 1742 letter to Euler, the computational verification by Oliveira e Silva (2013), and Helfgott's 2013 proof of the weak conjecture. Explain what Helfgott proved, what method he used (the circle method), and why the strong conjecture remains open.

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