Plasma AI

Last week we released the code and data behind Plasma researcher @IdaeanDaktyl’s proposed decipherment of Linear A. Here’s a closer look at what his readings of names and common words could reveal about Minoan Crete’s language, religion, and cultural ties.

Erdős #940: Which integers can you build by adding numbers whose prime factors all occur with high exponents? Call a number “r-powerful” if every prime in its factorization has exponent at least r. For example, 16 = 2⁴ and 2,592 = 2⁵ × 3⁴ are both 4-powerful. In 1976, Erdős asked whether infinitely many integers cannot be written as a sum of at most r r-powerful numbers, for each r ≥ 3. We proved the r = 4 case, and more: the exceptions occupy a positive proportion of the integers. Even with four summands and every 4-powerful number available, a persistent fraction remains unreachable.

Erdős 420, parts 2 and 3: How much does the number of divisors grow when you extend a factorial by just a few terms? In 1980, Erdős and Graham asked whether the ratio of divisor counts for (n + ⌊log n⌋)! and n! can get arbitrarily close to every number greater than 1. We proved it can. Pick a target—1.01, π, a million—and any desired accuracy. The ratio comes that close for arbitrarily large n. The same holds if you replace log n with any nondecreasing function that grows without bound but stays below log n, however slowly it grows.

@axiommathai’s CFSG formalization shows agent swarms can now carry a proof spread across 500 papers and 15,000 pages through the Lean kernel in 24 days. What excites us most is what gets built on top. Our Erdős 420 proof already stands on their PrimeGapsLib formalization of Maynard’s sieve, extended by our own agents. Libraries like these help people and agents build on each other’s work to advance research.

Tom is continuing this research with Fractal, Plasma’s platform for coordinating teams of cloud agents around shared knowledge. He is now applying it to other undeciphered scripts, bringing parallel investigations into a common research workspace.

On Monday we shared our proof of Erdős #111, posed by Erdős, Hajnal, and Szemerédi in 1979. Here’s more about what the result means and how the proof works. Start with a graph that cannot be properly colored even with a countably infinite list of colors: 1, 2, 3, ... For each n, let h(n) be the worst-case number of edges you must delete from an n-vertex piece to make it two-colorable, as shown in the video. We proved h(n)/n → ∞. Pick any deletion budget per vertex: a hundred, a million. For every sufficiently large n, there is an n-vertex piece that exceeds it.

Yesterday we shared our proof of Erdős #173, a geometry conjecture dating to 1973. The video shows the puzzle: color every point of the plane red or blue, and try to avoid a triangle of a chosen shape and size with all three corners the same color. You can avoid equilateral triangles of one particular size. But then equilateral triangles of every other size must appear with all three corners the same color. So must copies of every non-equilateral triangle, with exactly the prescribed shape and size.