AI models factor larger numbers, but costs grow exponentially

The authors report one successful factorization among 15,523 held-out 68-bit numbers using a large learned model and mathematical post-processing, while finding prohibitive scaling toward cryptographic sizes.

Big Tech

Brendan O Donoghue · Nenad Tomasev · Volodymyr Mnih · Timothy Nguyen · Dalin Guo · Arun Nair · +9 more

Google DeepMind

Research Digest·10 November 2026·2 min read

Thread:Compute-Efficient Scaling

Illustration: AI models factor larger numbers, but costs grow exponentially

In one line

A transformer using discrete diffusion and beam search factors 68-bit semiprimes but scales exponentially with bit length.

Before you read this

Three ideas this paper builds on

diffusion modelA type of generative model that learns to reverse a gradual noising process to create data.184 papers →scaling lawA predictable power-law relationship between compute or data and model performance.30 papers →beam searchA search algorithm that expands multiple promising candidates simultaneously to balance generation quality and computational cost.7 papers →

Read it as

The security of online communications depends on the difficulty of breaking a large number into its prime factors, especially a semiprime (product of a pair of primes). Traditional methods get slower as numbers grow.

A transformer AI model was trained with discrete denoising diffusion (learning to recover data from noise) and guided by (exploring many candidates) to predict factors from examples alone, without mathematical shortcuts.

The model factored 68-bit numbers, a new best for machine learning. But computing needs grow so fast with size that real-world encryption is not at risk. The approach remains far slower than classical factoring.

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Concepts

diffusion model scaling law beam search test-time scaling