Is there a probabilistic approach to number theory conjectures?

In summary, the conversation discusses using probabilistic approaches to study general conjectures in number theory. There is mention of using probabilistic models to make computational searches and examples such as Goldbach's hypothesis and the Erdos-Kac theorem are given. The use of probabilistic considerations in math is debated, with some mentioning historical examples and others arguing against it. The relationship between the Mertens conjecture and the Mobius function is also discussed, with an emphasis on the limitations of using a random walk model.
  • #1
arivero
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For instance, let's say that you want to study fermat x^n+y^n=z^n for n=3; do not mind that we already know the answer :-) We could consider the densities of exact cubes, d(n), and then to calculate joint probabilities for d(Z), d(X) and d(Y).

The mechanism can be applied, for instance, to decide where to look when doing computational searches. So, is there some branch of number theory studying such probabilistic approach for general conjectures?
 
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  • #2
Don't know if it's worthy to be called a whole branch, but yes. For example, we don't have proof of Goldbach's hypothesis, but we have results regarding the probabilities to write any number as a sum of primes that strongly indicate that it's true. There are assorted results based on a so called Cramer's probabilistic model of primes.
 
  • #3
probabilistic considerations are NOT VALID in math

for example , according to probabiltiy theory the MObius function is realted to a Random motion that may take both 1 and 0 with probability 1/2 so [tex] M(x) = o(x^{1/2+e}) [/tex] for any positive 'e' number this is JUST Riemann Hypothesis

another curious alternative i proposed to evaluate ASYMPTOTIC sums over primes was the following

since the probability of a number being prime is about [tex] 1/logx [/tex] then replacing the sum by a series and using this fact

[tex] \sum_{p\le T}f(p) \sim \int_{2}^{T}dx \frac{f(x)}{logx} [/tex]

for an smooth f(x)
 
  • #4
"probabilistic considerations are NOT VALID in math"

hyperbole not supported by history.
To the OP

Look at some of the work of Paul Erdos, Marc Kac, and Aurel Winter (from the 30s or 40s, I can't remember at the moment). For a more recent reference, "Introduction to Analytic and Probabilistic Number Theory" by G. Tenenbaum.
 
  • #5
The Erdos-Kac theorem is well-known, but what did Winter show? A different version of the same, or something else entirely?

(I'm having trouble finding anything. Google and mathscinet failed me.)
 
  • #6
Sorry - spelling error generated by poor coordination today. Wintner is the last name. They did some work on additive functions and their asymptotic distributions, and there is, I believe, an Erdos-Wintner theorem in this area as well.
 
  • #7
No problem. I've read^h^h^h^hskimmed some of Wintner's papers, but somehow the name didn't come to mind...
 
  • #8
^H! :smile: I haven't seen that in a while; strikeout seems to be [STRIKE]universal[/STRIKE] more common these days.
 
  • #9
according to probabiltiy theory the MObius function is realted to a Random motion that may take both 1 and 0 with probability 1/2

First of all, that's 1 and -1 with probability [itex]3/\pi^2[/itex]. Second of all, M(x) is only a random walk in the lowest approximation. If you do the computation more accurately, there's also a tendency to revert to the mean. If it were a random walk, we'd have a counterexample to Mertens conjecture long before 10^10^10.
 

Related to Is there a probabilistic approach to number theory conjectures?

1. What is a probabilistic approach to number theory conjectures?

A probabilistic approach to number theory conjectures involves using methods from probability theory to make predictions or estimations about the truth or validity of number theory conjectures. This approach is often used when traditional methods of proof are difficult or impossible to apply.

2. How is a probabilistic approach different from traditional methods of proving number theory conjectures?

Traditional methods of proof in number theory involve using logical arguments and mathematical principles to establish the truth of a conjecture. In contrast, a probabilistic approach relies on statistical analysis and probability theory to make educated guesses about the truth of a conjecture.

3. What are the advantages of using a probabilistic approach in number theory?

One advantage of using a probabilistic approach is that it can provide insights and predictions about conjectures that are difficult to prove using traditional methods. It can also help identify patterns and connections between different conjectures, leading to new discoveries and insights in number theory.

4. Are there any limitations to using a probabilistic approach in number theory?

Yes, there are limitations to using a probabilistic approach. This method can only provide probabilistic evidence or support for a conjecture, rather than a definitive proof. It also relies on assumptions about the distribution and behavior of numbers, which may not always hold true.

5. Can a probabilistic approach be applied to all number theory conjectures?

No, a probabilistic approach may not be applicable to all number theory conjectures. Some conjectures may be too complex or have too many variables to accurately apply probabilistic methods. Additionally, certain conjectures may require deterministic proof methods to establish their truth.

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