Five problems that have resisted every mathematician who touched them. Their human stories, their position on the prime ladder, and what that position reveals — and does not resolve.
The prime ladder — the sequence of bases LCM(1, 2, …, n) — was built in WP46 to answer a simple question: what do you find when you keep extending primes? The answer included a map: five open conjectures in number theory, each living at a different rung. WP46 placed them on the map. This paper tells the stories behind them.
These five problems share one quality: their statements require no advanced mathematics to understand, and their proofs have defeated every approach anyone has brought. The simplest and the hardest are not always different things.
7 June 1742. Christian Goldbach, a Prussian mathematician working as a tutor to the Russian royal family in St. Petersburg, wrote a letter to Leonhard Euler in Berlin. In the margin of that letter, Goldbach noted an observation: every integer greater than 2 seemed expressible as the sum of three primes.
Euler replied. He had a stronger version: every even integer greater than 2 is the sum of two primes. Not three — two. He considered it "a completely certain theorem" though he could not prove it. The phrase in Latin: certissimum esse theorema.
That exchange is the entirety of the conjecture's origin. Two mathematicians, a letter, a marginal observation, a stronger version offered in reply. Two hundred and eighty years have passed. Neither version has been proved.
The conjecture has been verified computationally for all even integers up to 4 × 10¹⁸. That is four quintillion individual cases, each checked. Every one of them decomposes as the sum of two primes. The pattern holds without exception. The proof does not exist.
The Hardy-Littlewood circle method — developed in the 1920s — gives the best analytic approach. It works over the complex numbers, through the zeta function and its relatives. It predicts, with high accuracy, how many ways a large even number decomposes into two primes. It does not prove the decomposition always exists.
Goldbach lives at every rung simultaneously. It is a statement about all primes — not about any particular prime's relationship to a base. Changing base does not illuminate it because its structure is analytic: it concerns the distribution of primes across all integers, not the termination properties of fractions in a given numeral system. This is the conjecture most immune to the ladder. It stands apart from the others here precisely because it does not belong to any rung. It belongs to the whole.
Take any positive integer. If it is even, divide it by two. If it is odd, multiply by three and add one. Repeat. The conjecture: no matter what number you begin with, you eventually reach 1.
Try it. Start at 6: 6, 3, 10, 5, 16, 8, 4, 2, 1. Start at 27: the sequence takes 111 steps to reach 1, passing through a peak of 9232. Start at any number. The conjecture says you always come down.
1937. Lothar Collatz, a German mathematician, proposed the problem. He had been collecting similar iterations for years, fascinated by the way simple rules produced unpredictable trajectories. He presented it at international conferences. It spread through the mathematical community the way simple problems do — by being impossible to ignore and impossible to solve.
Paul Erdős encountered it. He said: mathematics is not yet ready for such problems. He offered a cash prize for a proof. He never paid it.
The two operations split on a single distinction: even or odd. That distinction is a base-2 question — it reads the last bit of the binary representation. Dividing by 2 is a right-shift in binary. Multiplying by 3 is a ternary step. The conjecture mixes two different prime structures in every iteration.
Base 6 = LCM(1,2,3) = 2 × 3 is the rung of the ladder where both primes are simultaneously present in the base. Working in base 6, the even/odd split and the ×3 operation are both legible in the same numeral without conversion. Researchers have studied Collatz in base 3, base 6, and generalizations. The problem has not yielded to any of these lenses. What base 6 offers is clarity about where the two operations live — not a proof that they always compose toward 1.
April 2013. Yitang Zhang submitted a paper to the Annals of Mathematics. The title: "Bounded Gaps Between Primes." Zhang was 58 years old. He had spent years working as an accounting aide and teaching calculus at a university in New Hampshire that was not a major research institution. He had published almost nothing in two decades.
The Annals referees read the paper in three weeks and accepted it. This was extraordinary — the journal typically takes months to years. The referees had found a complete proof of something the mathematical community had considered far out of reach: that there exist infinitely many pairs of primes that differ by at most 70 million.
Seventy million is not two. But it was the first finite bound ever proved. Before Zhang, no one had shown that prime gaps were bounded at all — that the primes did not simply spread apart and keep spreading. Zhang showed they return. The bound has since been tightened, through the Polymath Project and work by James Maynard, to 246. The twin prime conjecture — that infinitely many pairs differ by exactly 2 — remains open.
All primes greater than 3 are coprime to 12. That means, in base 12, every prime above 3 ends in one of four digits: 1, 5, 7, or B (eleven). All twin prime pairs (p, p+2) with p > 3 have p ending in 5 or B in dozenal — because those are the only residue classes where p and p+2 are both coprime to 12. The structure of twin primes is visible in the dozenal cycle. It has not been turned into a proof.
The Rhind Mathematical Papyrus was written around 1550 BCE, copied from an earlier document dating perhaps to 1650 BCE. It is one of the oldest mathematical texts known. A significant portion of its content is a table: for each odd integer n from 3 to 101, how do you write 2/n as a sum of unit fractions?
Unit fractions — fractions of the form 1/k — were the Egyptian system for expressing all non-integer quantities. The Egyptians did not write 2/3 directly. They wrote 1/2 + 1/6. They did not write 2/7. They wrote 1/4 + 1/28. The Rhind Papyrus is a table of these decompositions, computed and recorded for practical use. Three and a half thousand years later, mathematicians are still working on related problems.
1948. Paul Erdős and Ernst Straus asked: for every integer n ≥ 2, can you write 4/n as the sum of exactly three unit fractions? Not two, not four — three. The conjecture says yes, always.
Erdős spent his life traveling between universities, living from a suitcase, offering cash prizes for the problems he could not solve. This was one of them. The prize was never collected.
In base 12, the fraction 4/n is written 0;4 × (12/n) dozenal whenever n divides 12. The six divisors of 12 — 1, 2, 3, 4, 6, 12 — each give an immediate three-unit-fraction solution, because the arithmetic closes. WP45 established why: base 12 = LCM(1,2,3,4) is the minimal base in which halves, thirds, and quarters all terminate. The Erdős-Straus conjecture is most tractable in this base for exactly the cases where the base is designed to handle them. The remaining cases — n not dividing 12 — require other methods. The conjecture has been verified computationally for very large n. No general proof exists.
August 2012. Shinichi Mochizuki posted four papers on the website of the Research Institute for Mathematical Sciences at Kyoto University. No press release. No announcement to colleagues. No conference presentation. The papers were simply there, totalling over five hundred pages, claiming to prove the ABC conjecture.
Mochizuki had developed the theory over many years in near-isolation. The framework — Inter-Universal Teichmüller Theory — restructured the relationship between multiplication and addition across different completions of the rational numbers. It was unlike anything else in number theory. Mathematicians around the world began trying to read it. Most found it inaccessible without years of prerequisite study that existed only in Mochizuki's own prior papers.
In 2018, Peter Scholze and Jakob Stix traveled to Kyoto to discuss the proof directly with Mochizuki. They left unconvinced, identifying what they believed was an irreparable gap in a key lemma. Mochizuki maintained the proof was correct. The disagreement has not been resolved. The conjecture remains open. A proof may or may not exist.
The ABC conjecture concerns three coprime positive integers a, b, c satisfying a + b = c. It bounds how large c can be relative to rad(abc) — the product of the distinct prime factors of a, b, and c combined.
The radical strips a number to its prime skeleton: take any integer, list its distinct prime factors, multiply them together. The radical of 12 = 2² × 3 is 2 × 3 = 6. The radical of 2520 = 2³ × 3² × 5 × 7 is 2 × 3 × 5 × 7 = 210. The radical is the rung of the prime ladder that a number touches — the set of primes it contains, each counted once.
The ABC conjecture says, in essence, that c cannot be much larger than its prime skeleton. A number's radical places a ceiling on how far the number can exceed what its primes alone would produce. This is a statement about what the ladder does not cover — the gap between a number and its primorial floor.
None of these five conjectures has been proved. None has been disproved. The prime ladder does not change this. What it offers is a way of reading each problem in the base where its structure is most legible — where the relevant primes are visible in the numeral system, where the operations involved close cleanly, where the pattern can be seen without it being a proof.
Goldbach is legible everywhere and illuminated nowhere in particular by a change of base. Collatz is legible in base 6, where both its operations are native. Twin Primes and Erdős-Straus are legible in base 12, where the residue structure of primes and the arithmetic of unit fractions both simplify. The ABC conjecture is legible in the structure of the ladder itself — in what the radical function does and does not capture.
A map is not a solution. But a problem placed on the right map is easier to carry.
Erdős died in 1996, mid-conference, between sessions at a mathematics meeting in Warsaw. He had published nearly 1500 papers — more than any other mathematician in history. The problems he left open, with and without cash prizes attached, remain the landmarks of combinatorics and number theory. He left no estate. Everything he owned fit in a suitcase.