Technical Review of Twin Prime Likelihood by Chat GPT 1. Use of Totient Function and Prime Frequencies The use of Euler's totient function (phi(a)) to determine the frequency of a prime p_n as the smallest factor in an interval is mathematically sound. The derivation of F_n = phi(P_(n-1)) / P_n is consistent with the properties of phi. However, the simplifications in the asymptotic formula require scrutiny: The approximation Prod(1 - 1/p_i, i=1 to n-1) ~ e^(-gamma) / log(p_(n-1)) is derived from Mertens' third theorem. While this is valid asymptotically, the paper does not explicitly address the error terms introduced by this approximation for finite n. This could affect the accuracy of subsequent results, particularly when estimating F_n for moderate values of n. 2. Decay Rate of F_n The decay rate F_n ~ e^(-gamma) / (p_n * log(p_n)) and its further approximation F_n ~ e^(-gamma) / (n * (log(n))^2) are consistent with established asymptotics. However, there are implicit assumptions about the density and distribution of primes that might not hold uniformly. For example: The use of p_n ~ n * log(n) is valid for large n, but deviations for small or moderate n could introduce significant inaccuracies in practical computations. A clearer distinction between asymptotic and exact behavior would strengthen the paper. 3. Error Term E_n The estimation of E_n via the integral Integral(1 / (x * (log(x))^2), x=n to infinity) ~ 1 / log(n) is a standard technique and is correctly executed. However: The paper assumes that E_n sufficiently quantifies the "unaccounted primes" in P_n ± 1, but this relies on the independence of prime divisors of P_n ± 1. This independence is not rigorously justified; there could be hidden correlations between the prime factors of P_n and those of P_n ± 1. A more detailed discussion on how E_n interacts with other terms, or a numerical validation, would add rigor. 4. Independence of P_n - 1 and P_n + 1 The argument assumes that P_n - 1 and P_n + 1 are "independent" in their factorization properties, particularly when estimating their likelihood of being prime. While this is a common heuristic in number theory, it lacks a rigorous foundation. For example: The structure of P_n as a primorial inherently ties P_n - 1 and P_n + 1 to the prime set {2, 3, ..., p_n}. This could introduce dependencies not accounted for in the analysis. 5. Conclusion on the Infinitude of Twin Prime Centers The conclusion that the set B of primorials serving as twin prime centers is infinite is speculative and hinges on the probabilistic argument that E_n -> 0 as n -> infinity. While the heuristic is reasonable: The proof implicitly assumes that the likelihood of P_n - 1 and P_n + 1 being prime increases without bound as n -> infinity. However, there is no direct evidence provided to exclude the possibility of structural obstructions or unexpected correlations at large n. This is particularly critical since twin primes are an open problem, and any claims about their infinitude should be carefully qualified. 6. Clarity and Rigor Some derivations are presented in an informal tone, which might obscure key assumptions or steps for a mathematically rigorous audience. For instance: When deriving the decay rate of F_n, the transition from finite products to asymptotics should explicitly state the domain of validity and the contribution of neglected terms. The claim that P_n ± 1 is prime "with increasing likelihood" as n -> infinity is heuristic at best and should be presented as such. --- Recommendations 1. Error Analysis: Include a more detailed examination of the error terms in the asymptotics, particularly those associated with F_n and E_n. 2. Numerical Validation: Supplement theoretical results with numerical simulations to validate the proposed likelihoods for moderate n. 3. Clarify Independence Assumptions: Address the potential dependencies between P_n - 1 and P_n + 1 in greater depth, even if only heuristically. 4. Qualification of Results: Clearly distinguish between rigorous proofs, heuristic arguments, and conjectures, particularly in the discussion of B's infinitude. --- While the paper provides an interesting heuristic approach to the likelihood of twin prime centers, its conclusions should be interpreted cautiously, as they rely heavily on unproven assumptions about the independence and distribution of primes.
I appreciate it but id like to know your thoughts as well . 1) for moderate n F_n can be calculated explicitly . the approx is for use in the integral . Point 5 has been addressed in the dependence and independent cases outlined in github . 6. Numeric validation would not be useful because this proof relies on the behavior of F_n as opposed to its actual values .Again F_n can be calculated explicitly given the explicit formula . From my own (possibly wrong) understanding , my proof addresses independence to satisfy borel-cantelli . I have independently run this proof through chatgpt 01 and i received similar responses . I would like to know your opinion as a human where you think this is lacking . I would lastly like to justify that this is not heuristics , but probabilistic which is rigorous on its own if all conditions are satisfied , which to my understanding , they are . Additionally thank you for reading and/or watching .
@@danieleid8528 Thank you for your response. I appreciate it. While I don’t yet fully understand the subject, I found your paper fascinating and potentially groundbreaking. To better grasp its strengths and weaknesses, I asked ChatGPT to analyze it and shared the response with you, thinking it might be useful. I must admit, I didn’t fully understand the AI’s feedback either, but I thought it was worth sharing with you. I wish you the best of luck and success with your work!
I don't really understand this but seems really exciting, hoping for more development on this
Technical Review of Twin Prime Likelihood by Chat GPT
1. Use of Totient Function and Prime Frequencies
The use of Euler's totient function (phi(a)) to determine the frequency of a prime p_n as the smallest factor in an interval is mathematically sound. The derivation of F_n = phi(P_(n-1)) / P_n is consistent with the properties of phi. However, the simplifications in the asymptotic formula require scrutiny:
The approximation Prod(1 - 1/p_i, i=1 to n-1) ~ e^(-gamma) / log(p_(n-1)) is derived from Mertens' third theorem. While this is valid asymptotically, the paper does not explicitly address the error terms introduced by this approximation for finite n. This could affect the accuracy of subsequent results, particularly when estimating F_n for moderate values of n.
2. Decay Rate of F_n
The decay rate F_n ~ e^(-gamma) / (p_n * log(p_n)) and its further approximation F_n ~ e^(-gamma) / (n * (log(n))^2) are consistent with established asymptotics. However, there are implicit assumptions about the density and distribution of primes that might not hold uniformly. For example:
The use of p_n ~ n * log(n) is valid for large n, but deviations for small or moderate n could introduce significant inaccuracies in practical computations. A clearer distinction between asymptotic and exact behavior would strengthen the paper.
3. Error Term E_n
The estimation of E_n via the integral Integral(1 / (x * (log(x))^2), x=n to infinity) ~ 1 / log(n) is a standard technique and is correctly executed. However:
The paper assumes that E_n sufficiently quantifies the "unaccounted primes" in P_n ± 1, but this relies on the independence of prime divisors of P_n ± 1. This independence is not rigorously justified; there could be hidden correlations between the prime factors of P_n and those of P_n ± 1.
A more detailed discussion on how E_n interacts with other terms, or a numerical validation, would add rigor.
4. Independence of P_n - 1 and P_n + 1
The argument assumes that P_n - 1 and P_n + 1 are "independent" in their factorization properties, particularly when estimating their likelihood of being prime. While this is a common heuristic in number theory, it lacks a rigorous foundation. For example:
The structure of P_n as a primorial inherently ties P_n - 1 and P_n + 1 to the prime set {2, 3, ..., p_n}. This could introduce dependencies not accounted for in the analysis.
5. Conclusion on the Infinitude of Twin Prime Centers
The conclusion that the set B of primorials serving as twin prime centers is infinite is speculative and hinges on the probabilistic argument that E_n -> 0 as n -> infinity. While the heuristic is reasonable:
The proof implicitly assumes that the likelihood of P_n - 1 and P_n + 1 being prime increases without bound as n -> infinity. However, there is no direct evidence provided to exclude the possibility of structural obstructions or unexpected correlations at large n.
This is particularly critical since twin primes are an open problem, and any claims about their infinitude should be carefully qualified.
6. Clarity and Rigor
Some derivations are presented in an informal tone, which might obscure key assumptions or steps for a mathematically rigorous audience. For instance:
When deriving the decay rate of F_n, the transition from finite products to asymptotics should explicitly state the domain of validity and the contribution of neglected terms.
The claim that P_n ± 1 is prime "with increasing likelihood" as n -> infinity is heuristic at best and should be presented as such.
---
Recommendations
1. Error Analysis: Include a more detailed examination of the error terms in the asymptotics, particularly those associated with F_n and E_n.
2. Numerical Validation: Supplement theoretical results with numerical simulations to validate the proposed likelihoods for moderate n.
3. Clarify Independence Assumptions: Address the potential dependencies between P_n - 1 and P_n + 1 in greater depth, even if only heuristically.
4. Qualification of Results: Clearly distinguish between rigorous proofs, heuristic arguments, and conjectures, particularly in the discussion of B's infinitude.
---
While the paper provides an interesting heuristic approach to the likelihood of twin prime centers, its conclusions should be interpreted cautiously, as they rely heavily on unproven assumptions about the independence and distribution of primes.
I appreciate it but id like to know your thoughts as well . 1) for moderate n F_n can be calculated explicitly . the approx is for use in the integral . Point 5 has been addressed in the dependence and independent cases outlined in github . 6. Numeric validation would not be useful because this proof relies on the behavior of F_n as opposed to its actual values .Again F_n can be calculated explicitly given the explicit formula . From my own (possibly wrong) understanding , my proof addresses independence to satisfy borel-cantelli . I have independently run this proof through chatgpt 01 and i received similar responses . I would like to know your opinion as a human where you think this is lacking . I would lastly like to justify that this is not heuristics , but probabilistic which is rigorous on its own if all conditions are satisfied , which to my understanding , they are . Additionally thank you for reading and/or watching .
@@danieleid8528 Thank you for your response. I appreciate it. While I don’t yet fully understand the subject, I found your paper fascinating and potentially groundbreaking. To better grasp its strengths and weaknesses, I asked ChatGPT to analyze it and shared the response with you, thinking it might be useful. I must admit, I didn’t fully understand the AI’s feedback either, but I thought it was worth sharing with you. I wish you the best of luck and success with your work!
@@danieleid8528 I hope that one day I will be able to understand something of this fascinating subject and fully appreciate the depth of your paper.
@ thanks for your input. The feedback is greatly appreciated(:
😊