The Mathematical Universe
Each galaxy is a field of mathematics. Each star is a problem. Constellations connect related ideas.
Number Theory Galaxy
Erdos Problem #728
#728How large can the gap k = a + b - n be, given that a!b! divides n!k!?
Erdos Problem #205
#205A number-theoretic problem resolved with a formal Lean proof by AI.
Erdos-Straus Conjecture
For every integer n >= 2, the fraction 4/n can be expressed as a sum of three unit fractions.
Erdos Conjecture on Arithmetic Progressions
If A is a set of positive integers whose reciprocals diverge, then A contains arithmetic progressions of every finite length.
Erdos-Kac Theorem
The number of distinct prime factors of n is approximately normally distributed.
Erdos-Ginzburg-Ziv Theorem
Among any 2n-1 integers, there exist n whose sum is divisible by n.
Erdos-Turan Conjecture on Additive Bases
If B is an additive basis of order 2, the representation function is unbounded.
Erdos Problem #457
#457A number-theoretic problem resolved autonomously by AI with formal verification.
Combinatorics Galaxy
Erdos Problem #125
#125A combinatorial problem on set systems proposed by Erdos.
Erdos-Ko-Rado Theorem
Maximum size of an intersecting family of r-element subsets of an n-element set.
Erdos-Szekeres Theorem
Any sequence of more than (r-1)(s-1) distinct reals contains a monotone subsequence of length r or s.
Erdos Discrepancy Problem
For any infinite +/-1 sequence, partial sums along arithmetic progressions are unbounded.
Erdos Problem #846
#846A combinatorial problem independently solved by two AI labs.
Erdos Problem #347
#347A combinatorial problem solved through extensive human-AI collaboration.
Erdos Problem #1051
#1051A combinatorial problem solved by the Aletheia AI system.