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GPT-5 Pro has recently solved two distinct and notoriously difficult math problems, showcasing a powerful new level of abstract reasoning. These aren’t just clever tricks; one solution challenges an IMO-level benchmark, while the other disproves a long-standing assumption in information theory.

It’s worth noting that its top competitors, like Google’s Gemini 2.5 Pro (in “Deep Think” mode) and Anthropic’s Claude 4.5+, have not yet been publicly tested on these specific problems.

Here’s a simple breakdown of what happened.

1. The Algebra Puzzle: Yu Tsumura’s 554th Problem

GPT-5-Pro solved, in just 15 minutes (without any internet search), the presentation problem known as “Yu Tsumura’s 554th Problem.”https://t.co/tKae6Vo0Kb

This is the first model to solve this task completely. I expect more such results soon — the model demonstrates a strong… pic.twitter.com/5ntk692BbY

— Bartosz Naskręcki (@nasqret) October 5, 2025

What is it? This is a problem from a collection by Yu Tsumura, roughly at the difficulty level of the International Mathematical Olympiad (IMO). The task is to prove that a specific mathematical group, defined by the rules governing its two generators, is “trivial” (meaning it’s the simplest possible group). Because of its concise wording, it has become a benchmark to test if an AI has reached high-level mathematical reasoning abilities.

What did GPT-5 Pro do? It became the first AI model to solve the problem. According to independent mathematicians who tested the model, GPT-5 Pro produced a complete proof in just 15 minutes, without any internet access.

Why it matters: This is a direct measure of progress. Just a couple mounths ago, a research paper titled “No LLM Solved Yu Tsumura’s 554th Problem” argued that current models lacked the capability for such tasks. GPT-5 Pro’s success demonstrates the incredibly rapid pace of advancement in AI’s reasoning skills.

2. The Information Theory Breakthrough: Disproving Majority Optimality

What is it? This problem, known as “NICD-with-erasures majority optimality,” comes from information theory. Imagine two people receive corrupted versions of the same signal. They each try to guess a function based on their partial data, with the goal of maximizing the chances they both guess the same thing. For a long time, experts believed the best strategy was the “majority function” (essentially, a democratic vote among the data points).

What did GPT-5 Pro do? It proved this long-standing belief wrong. Instead of solving for the best function, GPT-5 Pro found a specific counterexample—a different function that performs slightly but definitively better than the majority rule under certain conditions.

Here’s the counterexample it found for a specific setup (p=0.4, n=5): f(x) = sign(x_1 - 3x_2 + x_3 - x_4 + 3x_5)

This function achieved a score of 0.43024, beating the best majority function’s score of 0.42904.

Why it matters: This is a fundamental problem with huge practical applications. Finding optimal functions for signal recovery directly impacts how we design error-correcting codes for data storage, communication channels, and data recovery. By disproving the old assumption, GPT-5 Pro has opened up a new chapter for research in the field.

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