Large Language Models Cannot Tame Pi

Any number related to a person's life can be found in a certain digit position of the circumference ratio (π).

This is because pi (π) is an irrational number; that is, its decimal expansion neither terminates nor repeats, and its sequence of digits is random. Although there is currently no proof that pi's digits are completely random, mathematicians generally believe that the distribution of pi's digits follows no pattern or regularity, and every sequence of digits is theoretically possible to appear randomly at some position.

This means that no matter which number combination you choose, including a personal birthday, phone number, or important year, it can theoretically be found at some position in pi. These digits may appear once or multiple times within a long segment of pi, with the specific location determined by the random distribution of the digits.

This phenomenon is related to uniform distribution in mathematics, which suggests that the probability of all digits (0-9) appearing in any digit position of pi is theoretically equal. Therefore, any finite-length combination of digits is possible to appear at some position in pi.

The person who used pi in the most peculiar way must be Donald Ervin Knuth. In 1969, he began developing a computer language that allowed users to digitally typeset mathematical formulas. Knuth then spent ten years designing the TeX document typesetting system and used pi for the version numbering of its development. Starting from TeX3, the current version is TeX 3.141592653.

Humans have been attempting to find π for four thousand years, but even today, we only get closer to its actual value. The first to perform a rigorous calculation of π was the ancient Greek mathematician Archimedes (287–212 BC). He used the Pythagorean theorem to calculate the areas of a regular polygon inscribed in a circle and a regular polygon circumscribed about a circle. Since the actual area of the circle must lie between these two, the areas of these polygons provided the lower and upper bounds for the area of the circle. He understood that this would only yield an approximation of π, not its exact value. Using this method, Archimedes derived that π is between 3.1429 and 3.1408.

On August 19, 2021, the University of Applied Sciences of the Grisons in Switzerland calculated the most accurate value of pi to 62,831,853,071,796 digits. Interestingly, the marvelous AI Large Language Models cannot tame π. I asked seven different Large Language Models, and they all gave me different answers.

I inquired about four numbers related to me: "May I ask at which digit position 1026 appears in the digits of pi?" The result was that every LLM gave me a different answer. In past tests, a few LLMs would always provide the same answer, but this time there was no consensus whatsoever.

I first tried GPT and got the answer 6284. I then asked GPT to self-verify: "What is the four-digit sequence starting at position 6284 in the digits of pi?" The reply I received was 7590, not 1026.

I tried Grok, and the reply I received was 1639. A reverse check also did not yield 1026, but 5807.

I tried Le Chat (Mistral.ai), and the reply I received was 176451. A reverse check yielded 3141.

I tried Qwen2.5-Max, and the reply I received was 39. A reverse check yielded 7169.

I tried DeepSeek, and the reply I received was 8580. A reverse check yielded 3099.

I tried Tulu 3, and the reply I received was 2480. A reverse check yielded 3282.

I then tried Gemini, and the reply I received was 175319. For the reverse check, it did not tell me the answer but suggested I use a mathematical tool to calculate it.

I finally tried Claude, and it did not tell me the answer.

The reason these Large Language Models gave incorrect answers is that they attempted to write their own program to find the answer, but the program failed to run correctly. I asked the Pi-Search page, and it answered: "The string 1026 appears at position 14678. This string appears 20,130 times in the first 200 million digits of π." This should be the correct answer. My test was conducted on February 24, 2025. After informing GTP or Grok that a specific online mathematical tool could provide the answer, they indeed admitted the error and, following my suggestion, used the tool to find the correct answer.

With the evolution of Large Language Models, they may be able to provide the correct answer upon the first inquiry in the future.

Donald Ervin Knuth