| Mathematicians aim to expand what we know. As such, they spend most of their time thinking about how to prove new statements. So once they turn a conjecture into a theorem, you'd expect them to move on to a new problem. Their job, after all, is to keep adding new truths to the list.
But the value of a proof isn't just in moving a new statement to the "known" column. Mathematicians want to understand the "why" as deeply as they can. It's possible to convince a reader that each step in a proof follows logically from the previous one, even if the reader doesn't fully understand the deep mathematical connections at play. Reviewing such a proof, you'd miss out on the intangible wash of sudden understanding that gets mathematicians to the chalkboard each morning. When they talk about a "beautiful" proof, they mean one that sates their peculiar thirst for genuine comprehension.
And so, even after something's been proved, mathematicians will often seek alternative routes to the same truth, especially if they find the existing path unsatisfying or inelegant. Often a re-proof will illuminate its conclusion in ways that mathematicians previously missed. And sometimes in the course of constructing a second or third or fourth proof, a mathematician will invent new tricks that lead to unrelated and surprising discoveries. The history of math is full of examples where new proofs of already-proved statements can be credited with finally moving something into the "understood" column.
The virtues that make one proof better than another are deeply subjective and human. It's because of these aesthetic values that the field is as often compared to the arts as it is to the sciences. To a mathematician, a proof is a symphony, a work of art concerned not only with the final note, but with the revealing journey one must take to arrive there.
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Unsurprisingly, the most re-proved theorems in math are often the most famous, and the oldest. And few mathematical obsessions are older or more famous than prime numbers. The primes are especially concentrated toward the low end of the number line — there are 15 primes between zero and 50, but only four between 10,000 and 10,050. The prime number theorem is an equation that describes this distribution. Two mathematicians independently proved it in 1896. But over a century later, new proofs continue to appear. There are many reasons that's the case: It's a fundamental theorem; many other theorems rely on it; and it's a nice way to keep a number theorist's proof chops honed.
A common source of discomfort mathematicians feel about proofs is the involvement of computers. Modern machines can carry out tedious computational tasks at a scale beyond any mathematician's ability, but the proofs they produce often lack whatever ineffable qualities their human counterparts prize. Thomas Hales' 1998 computer-based proof of the Kepler conjecture (a question about how many spheres you can cram into a box) was so controversial that he had to spend six years coding every step and feeding it into a formal verification program to convince everyone it was valid. And mathematicians continue to eagerly await a pencil-and-paper proof of the four-color theorem (about whether it's possible to color countries on a map with four or fewer colors, so that no adjacent countries have the same color) — even though Wolfgang Haken proved it half a century ago with the help of 1,000 hours of computer time.
Quanta contributor Lyndie Chiou and I recently co-authored a story about a notable re-proof. The "ten martini" problem asks what happens when electrons living on a grid of atoms are placed in a magnetic field. The mathematician Marc Kac once offered 10 martinis to anyone who could prove that the electrons' energy values formed a fractal pattern. Mathematicians eventually figured out a proof, but it never sat right with one of its authors, Svetlana Jitomirskaya. The proof involved settling different cases using different techniques, giving it the feel of a patchwork quilt, constructed piecemeal. The mathematical truth felt profound enough that Jitomirskaya suspected there should be a more elegant proof that dealt with all the different cases at once.
Some mathematicians are so entranced by beautiful proofs that they believe in, and seek, a platonic ideal of every mathematical argument. Paul Erdős famously spoke of "The Book," a tome known only to God that contains all such perfect proofs. Just a couple years after he died, two mathematicians published their version, Proofs From THE BOOK. In 2018, they spoke with Quanta contributor Erica Klarreich about the compendium's first five editions, and about why certain proofs belonged in The Book. | 
