Claude Mythos solves the Erdős conjecture: AI and mathematics

A weekend to rewrite the history of mathematics

On May 26, 2026, Anthropic engineer Sholto Douglas shared news that immediately captured the attention of the scientific community. The model Claude Mythos had solved the unit distance conjecture formulated by Paul Erdős in 1946. The solution had been found "over the weekend", during a single weekend. Douglas described it as a «cute, simple proof» : an elegant demonstration, not a forced computation.

This news comes just days after another groundbreaking announcement. OpenAI had just disproven the same conjecture with a different approach. Therefore, two of the world's leading AI labs have produced analogous results, independently, on one of the most well-known open problems in combinatorial mathematics. The signal is unmistakable.

To check out the technical details of the original announcement, you can read the original source on The Decoder .

What the concept of "serious overhang" really means

Douglas used the expression «serious overhang» to describe the current situation. In physics, overhang indicates a suspended mass ready to fall. In the AI context, the term suggests that the capabilities of models already far exceed what is applied in production. In other words, there is a latent potential that is still largely unused.

This concept has direct implications for businesses. In fact, it means that the models available today — and those arriving in 2027 — can already tackle problems considered unsolvable until yesterday. However, most Italian SMEs have not yet structured internal processes capable of leveraging this level of formal reasoning.

According to the analyses of McKinsey on the Global AI Index , less than 30% of mid-sized companies have integrated advanced AI models into their core decision-making processes. Therefore, the gap between those who adopt and those who wait is widening rapidly.

From abstract to practical: formal reasoning in SMB applications

The Erdős conjecture belongs to pure mathematics. Yet, the skills a model demonstrates by solving that type of problem are the same ones that power high-value industrial applications. Specifically, these are abilities like formal proof, finding hidden patterns, and generating non-obvious solutions from complex constraints.

Here are a few areas where these skills turn into a real competitive edge for SMEs:

• Logistics and supply chain optimization: models with advanced reasoning can spot routing and storage setups that traditional algorithms just miss.
• Code generation and review: the ability to produce formal proofs directly shows up in the quality of the generated code and the reduction of structural bugs.
• Contract analysis and compliance: constraint reasoning is the secret sauce behind automatically making sense of tricky legal clauses.
• Dynamic pricing and revenue management: advanced models find the sweet spot for pricing in scenarios with loads of connected variables.

We at SHM Studio we work every day to turn these breakthroughs into practical solutions for the Italian market. So, we totally get the gap between the lab and the average SME.

The competition between Anthropic and OpenAI: what changes for tool users

The fact that two labs have solved the same problem independently is no coincidence. It's a sign of an intensifying technological race. OpenAI and Anthropic are both investing heavily in mathematical reasoning as a general intelligence benchmark.

However, for companies that need to choose which platform to adopt, this scenario introduces new questions. Which model offers greater reliability in structured reasoning? Which integrates better with existing systems? Which has a sustainable pricing for an SME with a limited budget?

According to Gartner , by 2027 over 60% of new enterprise applications will use AI models with multi-step reasoning capabilities. Consequently, the choice of supplier today will have structural impacts in the coming years. This is not a short-term reversible decision.

If you want a hand with comparing different tools, feel free to check out the digital consulting services by SHM Studio , or explore our resources on blog dedicated to digital innovation .

The construction site is still open: what we don't know yet

A critical approach is necessary. The news of the resolution of the Erdős conjecture by Claude Mythos is currently based on an informal statement from an Anthropic engineer. The proof has not yet undergone formal review by the mathematical community. Therefore, it is important to distinguish between a promising signal and a certified result.

Furthermore, the speed at which these announcements come one after another makes objective evaluation difficult. The risk for companies is twofold. On one hand, ignoring real developments due to excessive skepticism. On the other hand, adopting immature tools based on unverified announcements. Therefore, the most effective strategy is to monitor carefully, experiment in controlled environments, and scale only when results are measurable.

In this sense, a structured approach to artificial intelligence adoption is better than rushing into implementation just because of temporary hype.

Outlook for 2027: what to expect in the upcoming quarters

The developments of these weeks are accelerating an already visible trajectory. In the next 12-18 months, it is reasonable to expect AI models capable of autonomously tackling complex optimization problems in a business context. This includes multi-scenario financial planning, predictive demand management, and automatic generation of marketing strategies based on structured data.

For Italian SMEs, the operational implications are concrete. Firstly, those who invest now in internal training and AI tool integration will have a structural advantage. Secondly, those who build clean data flows and documented processes today will be able to leverage the capabilities of next-generation models without having to start from scratch.

Finally, the dimension of content and communication strategy should not be overlooked. Models with advanced reasoning will also change how companies produce B2B content, manage digital campaigns, and optimize organic presence. On these fronts, SEO activities , digital marketing and strategic copywriting will evolve significantly.

What to do now: three operational priorities for tech SMBs

Facing these developments, we at SHM Studio we suggest Italian small and medium-sized businesses focus on three priority areas.

• Internal AI capability audit: verify which business processes could benefit from models with advanced reasoning. Not all use cases require the same capabilities. Therefore, precise mapping avoids budget waste.
• Guided experimentation: kick off pilot projects on specific use cases, with success metrics defined beforehand. For example, test the automated generation of analytical reports or the review of contract documents.
• Competitive positioning: to use the tools of LinkedIn Ads and Google Ads to credibly communicate one's capacity for innovation. The market rewards companies that tangibly show they are keeping up with the times.

To explore these topics further or get a personalized consultation, you can contact the SHM Studio team directly.