In July 2026, an AI produced a proof of a mathematical conjecture that had remained open since the 1970s. That same week, state-of-the-art models could still get a multi-digit multiplication wrong. This contradiction is not an anomaly—it follows directly from how these systems are built. And understanding it changes how you use them.
We reported in a recent article how an AI produced a mathematical proof that had resisted generations of researchers. How can a system capable of that fail at a calculation a ten-year-old can do in a notebook? Here is the explanation, in three layers.
First layer: it doesn't calculate
The fundamental point, already covered in our article on LLMs: a language model predicts the most likely next piece of text. It does not perform arithmetic operations.
When you ask it what 7 times 8 is, it doesn't multiply. It generates the text that most plausibly follows that question, and since "56" appears millions of times after that phrasing in its corpus, it answers correctly. That's not calculation—it's pattern recognition.
This distinction explains everything else. For common operations, widely present in training texts, pattern recognition works perfectly. For a multiplication of two random four-digit numbers, there is no memorised pattern: that exact operation has likely never been written in its corpus. The model must then improvise, and it improvises badly.
Second layer: numbers are poorly segmented
Here is the technical explanation few people know, and it's an elegant one. As we saw in our article on tokenizers, an AI doesn't read characters but tokens—those fragments learned according to their frequency.
Numbers undergo this segmentation just like the rest of the text. A number like 1,234 can be split into two or three fragments, based on groupings that have nothing to do with its mathematical value. The units digit, the tens digit, the hundreds digit can end up scattered across different tokens.
Imagine being asked to add two numbers, but instead of seeing their digits aligned, you receive arbitrary bundles of symbols. You know how to add, but the information arrives in a format that prevents you from setting out the operation. That's exactly the model's situation. Arithmetic relies on digit position, and token segmentation destroys part of that positional information.
Here's the counter-intuitive twist. A mathematical proof is a chain of reasoning expressed in language: stating a hypothesis, applying a theorem, deriving a consequence. It's structured text—exactly what these models handle best. A multiplication, by contrast, is a mechanical manipulation of digit positions, which is what they do worst. From a language model's perspective, high-level mathematics is closer to philosophy than to mental arithmetic.
Third layer: what has changed
Recent models make far fewer mistakes than two years ago, for three reasons.
Explicit reasoning first. As we explained in our dedicated article, a model that breaks an operation down step by step on its internal scratchpad, like setting out a multiplication on paper, does far better than one that tries to produce the result in a single go. It reconstructs what token segmentation had scattered.
Tools next. The most reliable solution is not to ask the model to calculate, but to give it access to a calculator or a code interpreter. It then writes the operation and lets a deterministic machine execute it. This is exactly the logic of the intermediary tool we described regarding AI in biology: rather than making the model better, you hand it the right instrument.
Targeted training finally, with tokenizers that handle digits more cleanly and datasets specifically designed for arithmetic.
What to take away
This quirk is an excellent reminder of what these systems really are. They are not miniature brains that can do everything a little. They are language-manipulation machines, extraordinarily gifted at anything expressed in words and reasoning, and structurally clumsy at anything requiring precise mechanical handling of positioned symbols.
The practical reflex follows: for any calculation that matters, insist that the AI use a calculation tool or spell out its steps, and verify the result. Not because it's stupid, but because you're asking it to do precisely the thing it's least designed for. It's a bit like asking an excellent speaker to do your mental arithmetic: they might manage, but that's not where their talent lies.