The Obviously True Theorem No One Can Prove

In winter 1954, Xiamen, China is under bombardment as the People's Republic of China attempts to take control of the nearby Kinmen Islands from anti-communist forces, with artillery shells raining down on both sides.

21-year-old Chen Jingrun spends his time in an air raid shelter reading math books and working on solving one of the oldest problems in mathematics, rather than waiting in fear like others around him.

The central question: Can every even number greater than 2 be written as the sum of two primes? A problem so simple a child can understand it, yet unsolved by the greatest mathematicians for nearly 300 years.

Chen's high school teacher told the class that number theory is the queen's crown and Goldbach's conjecture is the pearl on the crown. Chen was the only student who didn't laugh, secretly vowing to solve it.

• 6 = 3 + 3 • 10 = 5 + 5 or 7 + 3 • 42 = 37 + 5

Using a pyramid of prime numbers with diagonal lines, where each sum of two primes creates an even number. As you go down the pyramid, even numbers appear more frequently and seem to always occur.

Known as Goldbach's conjecture, stating that every even number greater than 2 can be written as the sum of two primes, and these representations become more frequent for larger numbers.

Christian Goldbach was a little-known Prussian mathematician from the early 1700s who traveled across Europe meeting great mathematicians including Leibniz, Newton, and Bernoulli before settling in Russia.

In 1727, Goldbach met the 20-year-old math prodigy Leonard Euler at the St. Petersburg Academy of Sciences. The two quickly bonded over a shared obsession with number theory and corresponded for 35 years.

On June 7, 1742, Goldbach wrote to Euler: 'It seems that every integer greater than 2 can be written as the sum of three primes.' This was not exactly the form of the conjecture that now bears his name.

Euler thought the idea could be refined further. He realized that odd numbers might need three primes, but for even numbers, just two primes would be enough to sum to every even number.

• Weak Goldbach: Every odd number greater than 5 can be written as the sum of three primes • Strong Goldbach: Every even number greater than 2 can be written as the sum of two primes • The strong conjecture implies the weak one, but not vice versa

In 1923, G.H. Hardy and John Littlewood used the prime number theorem to create a function estimating how many ways an even number can be written as the sum of two primes.

The prime number theorem states that a large number in the neighborhood of N has about 1 over the natural logarithm of N chance of being prime, allowing probability-based estimates.

Their calculations showed that the expected number of ways to write a number as the sum of two primes increases for larger numbers, appearing as approximately N over ln(N) squared.

Despite their work, Hardy and Littlewood could only provide an estimate, not a proof. As they noted in their conclusion, 'It is only proof that counts.'

In 1913, Hardy received a 10-page letter from an unknown mathematician, Srinivasa Ramanujan from India, containing over 100 theorems with little explanation but remarkable mathematical insights.

Some of Ramanujan's formulas were already known, others completely new, but some seemed impossible. Hardy and Littlewood debated whether he was a genius or a fraud before recognizing his phenomenal talent.

Ramanujan worked on intuition and claimed his personal goddess Namagiri would come to him in dreams and write formulas on his tongue. He said an equation has no meaning unless it expresses a thought of God.

Around 1917, Hardy and Ramanujan together invented the circle method, used to tackle different problems in number theory. This would become the main method to address the weak Goldbach conjecture for the next hundred years.

Hardy and Littlewood imagined a machine that takes all possible combinations of three primes smaller than N, adds them up, and counts how many equal N.

The method uses exponential functions and integrals to create a function that returns 1 when three primes sum to N, and 0 otherwise, effectively building a mathematical counting machine.

• For N=10,001 with 1,229 primes, there are 310 million possible triplet combinations • For N=1 billion, combinations jump to nearly 22 sextillion • Brute force approach becomes computationally impossible

Hardy and Littlewood reframed the problem by moving the sum inside the integral, analyzing the collective behavior of all primes at once rather than checking combinations one by one.

Each prime number is represented as a clock that spins at a rate determined by that prime. When adding primes, the clocks are added tip to tail, and their interference patterns reveal important structure.

• At certain values like alpha = 1/6, 1/3, and 1/2, the clocks align constructively • All clocks point in similar directions, creating a large resultant value • These correspond to specific rational fractions in the complex plane

When dividing primes by rational numbers, they create restricted remainders. For example, when dividing by 2, only two remainders (0 or 1) are possible, causing most primes to interfere constructively.

The majority of contributions come from small regions called major arcs. The rest of the circle provides minor arc contributions. For weak Goldbach, the main term grows faster than the error term.

In 1937, Russian mathematician Ivan Vinogradov proved the same result as Hardy and Littlewood but without assuming the generalized Riemann hypothesis, making his proof assumption-free.

Vinogradov proved the weak Goldbach conjecture held for sufficiently large numbers but didn't specify what that threshold was, leaving the result unsatisfying.

• One of Vinogradov's students specified the threshold at approximately 10 to the 6.8 million • By 1989, the bound was reduced to 10 to the 43,000 • By 2002, it dropped to 10 to the 1,346

Even at 10 to the 1,346, this number exceeds the number of protons in the universe (10 to the 80). Verification by computer is absolutely hopeless.

Peruvian mathematician Harald Helfgott, around 2005, believed he could improve on existing methods and spent the next eight years attacking the problem from two sides.

• Pushed computational verification as high as possible: 8.8 times 10 to the 30 • Refined mathematical approach to reduce the constant factor K • Combined both efforts to bridge the gap

By 2013, Helfgott brought K down to 10 to the 27, which was below the numbers his computers had already checked, proving the weak Goldbach conjecture true.

Helfgott published 'The Ternary Goldbach Conjecture is True,' proving that every odd number greater than 5 can be written as the sum of three primes, nearly 300 years after Goldbach's original letter.

In 1956, Chen Jingrun was recognized for his mathematical prowess and became an assistant at the Chinese Academy of Sciences, dedicating the next 10 years to number theory including Goldbach's conjecture.

By 1966, Chen proved that every sufficiently large even number is the sum of a prime and a semi-prime (either a prime or the product of exactly two primes).

This was the closest anyone had gotten to solving the strong Goldbach conjecture. Mathematician Andre Weil compared following his proof to climbing along the top of the Himalayas in the stratosphere.

Despite severe persecution during the Cultural Revolution (1966-1976), including torture and forced labor, Chen secretly worked on his math by kerosene lamp light and published his theorem in April 1973.

In May 1966, Mao Zedong declared the Cultural Revolution to purge bourgeois elements. The Red Guard stormed institutions, burned books, and persecuted intellectuals including scientists and professors.

Chen was forced into manual labor and made to live in a converted boiler room with no electricity. The Red Guard insulted, spat on, and beat him so severely he lost consciousness repeatedly.

Despite the brutality, Chen kept secretly working on his math under dim kerosene lamp light. One report suggested he may have attempted suicide by jumping from a third-story building, landing on the second floor.

After Mao died in 1976 and the Cultural Revolution ended, journalist Xu Chi wrote about Chen's achievements and hardship. He became a national hero, celebrated in schools, books, films, and had an asteroid named after him in 1996.

For the weak Goldbach conjecture, the circle method works because the main term grows faster than the error term. For the strong conjecture, the major arcs are no longer major, and the main contribution comes from the minor arcs.

To solve the strong conjecture, a fundamentally new technique or approach is needed. Despite centuries of effort, mathematicians haven't found the right method yet.

• In 1938, Nils Pipping checked all numbers up to 100,000 by hand without finding a counterexample • Modern computers have verified the conjecture up to four quintillion • The pattern known as Goldbach's comet shows more ways exist for larger numbers

Either there's a conspiracy where the conjecture breaks down at some very large number, or it's true throughout all numbers. The consistent pattern in the data and lack of drops suggest it should be true.

Goldbach's conjecture doesn't have direct applications to the real world and doesn't seem to affect other areas of mathematics, raising the question of why mathematicians bother solving it.

The Goldbach conjecture might be opening up a whole new part of the multiverse of mathematical ideas. We don't have the bird's eye view of mathematics to know what's truly central versus peripheral.

People should do what fires them up because passion leads to remarkable discoveries. Working on something just because you think it's important tends to produce second-rate results.

The modest view is that we don't know what's necessarily important, but we do know what we love. Work on what you love, and let your passion drive remarkable achievements.

Goldbach's conjecture frustrated mathematicians for centuries. For a long time it seemed unsolvable, and many gave up thinking it was too big to tackle.

It only took a few incredibly determined individuals who rejected the status quo to keep pushing toward a solution. Chen Jingrun's obsession, begun during wartime bombings, exemplifies this determination.

Life is full of inconveniences that people accept as unsolvable. Shaving irritation, razor burn, and constant razor replacement were once problems Derek accepted until finding the Henson Razor.

The Henson Razor was developed by aerospace experts who applied obsessive focus to create a better product. CNC machines achieve tight tolerances with blade extension of 0.03 millimeters maintaining a precise 30-degree cutting angle.