#2993 Applying Generative-AI to Identify New Mathematical Theorems and Scientific Principles

Google’s new AI generates hypotheses for researchers

https://arstechnica.com/google/2025/02/googles-new-ai-generates-hypotheses-for-researchers/

Googles New AI Co Scientist Just Changed EVERYTHING (AI co-scientist Explained)

https://youtu.be/P4V8zFaJsu0

Simulating scientists: A new tool for AI-powered scientific discovery

https://techxplore.com/news/2025-02-simulating-scientists-tool-ai-powered.html

LLM4SD: https://github.com/zyzisastudyreallyhardguy/LLM4SD

#2993 Describe how I can request ChatGPT with GPT 4 to produce new mathematical theorems and/or scientific principles for a user-specified problem statement

To request ChatGPT with GPT-4 to generate new mathematical theorems or scientific principles for a user-specified problem statement, follow these structured steps:

1. Define the Problem Statement Clearly

Start by providing a precise problem statement, including:
- The field of mathematics or science (e.g., algebra, topology, physics, thermodynamics).
- The constraints or assumptions.
- Existing related theorems, principles, or formulas.

Example Prompt:
"I want to generate a new mathematical theorem related to prime number distribution. Assume the Riemann Hypothesis is true. How can we formulate a new conjecture about prime gaps?"

2. Request Theorem Generation Using Known Techniques

Ask ChatGPT to examine existing methods and suggest extensions, such as:

Generalization of existing theorems.
Axiomatic derivation.
Computational evidence leading to a conjecture.
Applying abstract algebra, group theory, or topology.

Example Prompt:
"Using Fourier analysis on prime-counting functions, derive a potential theorem that could extend the Prime Number Theorem."

3. Request New Scientific Principles

If working on scientific principles, guide the AI to explore:

Axioms and fundamental assumptions.
Empirical observations.
Thought experiments.
Alternative mathematical formulations.

Example Prompt:
"Propose a new principle in thermodynamics that accounts for entropy decrease in self-organizing systems, considering non-equilibrium statistical mechanics."

4. Request Proof Sketches or Justifications

Once a theorem or principle is suggested, ask for:

Logical proof outlines.
Computational verifications.
Experimental designs to validate it.

Example Prompt:
"Provide a proof sketch for a proposed theorem that refines the distribution of twin primes using probabilistic number theory."

5. Request Iterative Refinement

After generating an initial response, ask for modifications by:

Strengthening assumptions.
Finding counterexamples.
Providing alternative formulations.

Example Prompt:
"Can this theorem be extended to transcendental number distributions? If not, what counterexamples exist?"

6. Translate into Formal Notation (LaTeX)

Once satisfied with the formulation, request the theorem in LaTeX format for formal representation.

Example Prompt:
"Write the proposed theorem and its proof in LaTeX format."

Example Application

Prompt:

"Propose a new combinatorial theorem that strengthens the Erdős–Ko–Rado theorem in high-dimensional hypergraphs."

Expected ChatGPT Response:

Statement of a conjecture.
Possible proof outline or computational justification.
Suggestions for further exploration.

Final Thought

ChatGPT does not guarantee mathematical correctness, but it can generate novel ideas that require rigorous human verification. Using symbolic computation tools (like Wolfram Alpha, Coq, or Lean) in parallel can help verify and refine AI-generated results.

Would you like an example theorem formulated?

Yes, let's start with prime number sequences where each interval has it's own characteristics, and the sequence of intervals have a consistent pattern

Let's define a new theorem about prime number sequences where each interval exhibits unique characteristics, yet the sequence of intervals follows a consistent pattern.

Problem Statement

We seek a pattern in prime number sequences where each interval $[n_k, n_{k+1}]$ has distinct characteristics, and the sequence of these intervals follows a predictable rule.

Observations:

Prime gaps are irregular but can be analyzed statistically.
Existing results include:
- The Prime Number Theorem approximates the number of primes less than $x$ as $\frac{x}{\ln x}$ .
- Cramér’s conjecture suggests $g(n) = O((\ln n)^2)$ for the prime gap function $g(n) = p_{n+1} - p_n$ .
- The Hardy-Littlewood k-tuple conjecture estimates the distribution of prime gaps.

We propose a structured sequence of prime intervals where the characteristics within each interval form a predictable pattern.

Proposed Theorem (Conjecture)

Prime Interval Pattern Theorem (PIPT):
There exists an infinite sequence of prime intervals $I_k = [P_k, P_{k+1}]$ such that within each interval:

The average prime gap $\bar{g}(I_k)$ follows a smooth function $f(k)$ .
The density of primes within $I_k$ follows a consistent pattern given by $D(I_k) \approx \frac{c}{\ln P_k}$ , where $c$ is a sequence-dependent constant.
The local distribution of prime residues modulo small integers exhibits a near-periodic structure.

Further, the sequence $\{I_k\}$ satisfies:

A recurrence relation $I_{k+1} = f(I_k)$ for a predictable function $f$ .
A scaling property: If $I_k = [P_k, P_{k+1}]$ , then $∣ I_{k + 1} ∣ \approx g (k) ∣ I_{k} ∣ for some function$ $g(k)$ .

Sketch of a Proof (or Justification)

Step 1: Define a Sequence of Intervals

Let $P_k$ be a starting prime for each interval.
Define $I_{k} = [P_{k}, P_{k + 1}] where$ $P_{k+1}$ is the first prime exceeding $P_k + f(k)$ , where $f(k)$ is an increasing function.

Step 2: Establish Predictability

We analyze:

Average prime gap:
$\bar{g}(I_k) = \frac{1}{|I_k|} \sum_{P \in I_k} (P' - P)$
and show that it approximates a function $g(k)$ .
Density function:
$D(I_k) = \frac{\pi(P_{k+1}) - \pi(P_k)}{|I_k|}$
We approximate this using the Prime Number Theorem.
Recurrence Relation:
$P_{k+1} = P_k + \Theta((\ln P_k)^c)$
for some constant $c$ , implying a structured pattern.

Step 3: Verify Experimental Evidence

Using computational methods, we:

Identify intervals $I_k$ satisfying the theorem.
Verify whether $D(I_k)$ and $\overset{ˉ}{g} (I_{k}) maintain consistency.$

Applications and Future Work

Prime Prediction: This can improve models for prime distributions in large intervals.
Cryptography: Understanding structured prime gaps aids cryptographic key generation.
Analytic Number Theory: Extends classical results on prime gap distributions.

Next Steps

Would you like a computational experiment to test this theorem? I can generate prime sequences and visualize their gap distributions.

Yes

The analysis of prime intervals is now available in the displayed table, showing key characteristics like average prime gap and density across different intervals. Additionally, the plots visualize:

Average Prime Gap vs. Start Prime
- This shows how the average gap between consecutive primes grows as the start prime increases.
- The log-log scale helps reveal any consistent patterns.
Prime Density vs. Start Prime
- Illustrates how prime density decreases as numbers grow larger.
- Again, using a log scale highlights underlying trends.

If you would like further refinements, such as testing alternative interval definitions or deeper statistical analysis, let me know!

import numpy as np
import sympy as sp
import pandas as pd
import matplotlib.pyplot as plt
import ace_tools as tools

# Generate prime numbers up to a certain limit
prime_limit = 100000 # Set an upper bound for prime number generation
primes = list(sp.primerange(2, prime_limit))

# Compute prime intervals and characteristics
intervals = []
average_gaps = []
densities = []
start_primes = []

# Define interval step size (logarithmic growth)
for i in range(1, len(primes) - 1):
P_k = primes[i]
P_k1 = primes[i + int(np.log(P_k))] if (i + int(np.log(P_k))) < len(primes) else primes[-1]

interval = (P_k, P_k1)
interval_primes = [p for p in primes if P_k <= p < P_k1]

if len(interval_primes) > 1:
gaps = [interval_primes[j+1] - interval_primes[j] for j in range(len(interval_primes)-1)]
avg_gap = np.mean(gaps) if gaps else 0
density = len(interval_primes) / (P_k1 - P_k)

start_primes.append(P_k)
intervals.append(interval)
average_gaps.append(avg_gap)
densities.append(density)

# Create DataFrame for visualization
df_intervals = pd.DataFrame({
"Start Prime": start_primes,
"End Prime": [i[1] for i in intervals],
"Avg Gap": average_gaps,
"Density": densities
})

# Display the table of prime intervals with characteristics
tools.display_dataframe_to_user(name="Prime Interval Characteristics", dataframe=df_intervals)

# Plot Average Prime Gap vs. Start Prime
plt.figure(figsize=(10, 5))
plt.scatter(start_primes, average_gaps, s=10, alpha=0.7)
plt.xlabel("Start Prime of Interval")
plt.ylabel("Average Prime Gap")
plt.title("Average Prime Gap vs. Start Prime")
plt.xscale("log")
plt.yscale("log")
plt.grid(True)
plt.show()

# Plot Density vs. Start Prime
plt.figure(figsize=(10, 5))
plt.scatter(start_primes, densities, s=10, alpha=0.7, color="red")
plt.xlabel("Start Prime of Interval")
plt.ylabel("Density of Primes in Interval")
plt.title("Prime Density vs. Start Prime")
plt.xscale("log")
plt.yscale("log")
plt.grid(True)
plt.show()

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