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On the impossibility of discovering a formula for primes using AI
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The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erd\H{o}s-Kac Law and Hardy-Ramanujan theorem in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.
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Cited by 2 Pith papers
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On arithmetic terms expressing the prime-counting function and the n-th prime
The paper constructs, in principle, fixed-length arithmetic terms for the prime-counting function pi(n) and the n-th prime p(n), with p(n) expressed as a hypercube-derived count of solutions to a 42-variable exponenti...
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Testing Transformer Learnability on the Arithmetic Sequence of Rooted Trees
A GPT-2 trained on the first 10^11 integers encoded as rooted-tree Dyck words reaches ~0.4 next-word accuracy, but the body does not contain the claimed controls or far-range test blocks.
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