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 he had solved the unit distance conjecture formulated by Paul Erdős in 1946. The solution had been found «over the weekend.» Douglas described it as a «cute, simple proof»An elegant demonstration, not a computational brute force.

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

To explore the technical details of the original announcement, it is possible to consult The original source on The Decoder.

What the «serious overhang» concept really says»

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 the models already far exceed what is currently deployed in production. In other words, there is a latent potential that remains largely untapped.

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 recently. However, most Italian SMEs have not yet structured internal processes capable of exploiting this level of formal reasoning.

According to the analysis of McKinsey on the Global AI Index, less than 30% of medium-sized companies have integrated advanced AI models into their core decision-making processes. Consequently, the gap between early adopters and those who are waiting is widening rapidly.

From abstract to concrete: formal reasoning in SME 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 capabilities include formal theorem proving, the search for hidden patterns, and the generation of non-obvious solutions from complex constraints.

Here are some areas where these capabilities translate into a concrete competitive advantage for SMEs:

• Logistics and supply chain optimization: Models with advanced reasoning can identify routing and storage configurations that classic algorithms do not explore.
• Code generation and review: The ability to produce formal proofs is directly reflected in the quality of the generated code and the reduction of structural bugs.
• Contract review and compliance: Constraint reasoning is the basis for the automated interpretation of complex legal clauses.
• Dynamic pricing and revenue management: Advanced models find optimal price equilibria in scenarios with many interdependent variables.

We of SHM Studio works daily to translate these advancements into operational solutions for the Italian market. Therefore, we understand the gap between the laboratory and the average SME very well.

The competition between Anthropic and OpenAI: what changes for those choosing tools

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

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 sustainable pricing for an SME with a limited budget?

According to Gartner, By 2027, more than 60% of new enterprise applications will use AI models with multi-step reasoning capabilities. Consequently, the choice of vendor made today will have structural impacts in the coming years. This is not a decision that can be reversed in the short term.

Those who wish for support in the comparative evaluation of the tools can refer to the SHM Studio digital consulting services, or explore our resources on digital innovation blog.

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

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

Furthermore, the speed at which these announcements follow one another makes objective evaluation difficult. The risk for companies is twofold. On the 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’adoption of artificial intelligence it is preferable to a rush to implementation driven by the enthusiasm of the moment.

Outlook for 2027: what to expect in the coming quarters

The developments of recent weeks accelerate a trajectory that is already visible. Over the next 12-18 months, it is reasonable to expect AI models capable of autonomously tackling complex optimization problems in the business environment. This includes multi-scenario financial planning, predictive demand management, and the automated generation of marketing strategies based on structured data.

For Italian SMEs, the operational implications are concrete. First, those who invest now in internal training and the integration of AI tools will have a structural advantage. Second, 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 size of the 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 e strategic copywriting will evolve significantly.

What to do now: three operational priorities for tech SMEs

Faced with these developments, we at SHM Studio We suggest Italian SMEs focus on three priority areas.

• Internal AI capabilities 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: launch pilot projects on limited use cases, with success metrics defined in advance. For example, testing the automatic generation of analytical reports or the review of contractual documents.
• Competitive positioning: use the tools LinkedIn Ads e Google Ads to credibly communicate one's innovation capacity. The market rewards companies that concretely demonstrate they are keeping up with the times.

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