The Prime Random Walk; can you prove it?
In our recent paper "Prime numbers and random walks in a square grid" published in Phys. Rev. E
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.104.054114
we conjectured that the area of the Prime Random Walk keeps growing with the number of steps indefinitely.
Can you prove or disprove it?
We (A. Fraile, D. Fernández) offer 1000 $ to the person who proves it right, or wrong.
* The proof has to be published in a peer reviewed well respected international journal to ensure its validity.
** Offer expires in 2 years counting from the date of publication of this offer (18-4-2024), so, the 18-4-2026.
*** The proof must be complete, i.e. not based of some unproved conjectures or hypotheses.
Alberto Fraile
0000-0003-2646-5159
Daniel Fernández
0000-0002-6896-1006
The conjecture about its area growing indefinitely offers a fascinating challenge. I look forward to seeing the developments in this area.
Dear Peter,
The walk is constructed from a sequence of steps dictated by and dependent on the sequence of the last digits of the primes.
The walk moves (in a square grid), up, down, left, right depending on the last digit of the prime numbers, since that last digit can be only 1, 3, 7 and 9 (except 2 and 5 that appear only once each and we ignore them).
The details can be found in the paper:
Prime numbers and random walks in a square grid. Phys. Rev. E 104, 054114 (2021)
More about the sequence itself here:
https://oeis.org/A007652
Thanks Neil for adding our paper to the OEIS.
Dear Peter,
Yes, the gaps are part of the problem.
but it is not only the gaps what matters.
It is the sequence itself what dictates the area (number of visited points without repetition).
We did not study gaps between primes in our paper. We just conjectured that the area of the Prime Walk keeps growing with the number of steps indefinitely.
is likely that the linear growth we observed up to N = 10^10 (Fig 2 in PRE)
will became "less linear" and the growth will decrease,
but we think it will continue growing.
As you said, gaps (and the probabilities of these gaps) for sure can tell something about the behavior of the prime walk, maybe that can provide some limits of the growth etc,
but, how can one prove that the area is limited or unlimited?
Dear Peter, Thank you for your insightful comments.
Indeed, you've accurately summarized the mechanisms of the Prime Walk, although the focus of our research is not solely on the distribution of steps or the prime gaps but rather on the spatial exploration on the grid as influenced by these steps. The "area" in our context refers to the number of distinct grid points visited by the walk. As you pointed out, one would expect that the number of distinct points visited by a typical random walk would grow indefinitely. However, the key aspect we're investigating is the growth rate and the nature of this growth. Is it linear, sublinear, or something else? The peculiar behavior in our Prime Walk compared to a traditional random walk arises from the deterministic nature of prime endings in base 10 influencing the directions taken. This deterministic aspect influences the walk differently compared to purely stochastic random walks. We have not yet compared the prime walk against predictions from the Hardy-Littlewood k-tuples conjecture, as our main conjecture was primarily focused on the growth behavior of the visited area. However, your points could potentially lead to a deeper understanding of the dynamics at play and what truly distinguishes prime walks from conventional random walks. Thank you again for your thoughtful engagement.
It is worth mentioning that the gaps between primes govern the 3D shape of the construct, since they dictate the z_max values. - by construction the values of Z_max in every point (x,y) are sums of gaps between primes.
And it seems that these z_max values clearly follow Benford’s law [1] as shown in Figure 6 of our paper [2]. (a proof was beyond the scope of the paper).
[1] F. Benford, The law of anomalous numbers, Proc. Am. Philos.
Soc. 78, 551 (1938).
[2] A. Fraile, O. Kinouchi, P. Dwivedi, R. Martínez, T. E. Raptis, and D. Fernández. Prime numbers and random walks in a square grid. Phys. Rev. E 104, 054114 (2021)
Remember, remember... 22 months left.
The Prime Random Walk; can you prove it?
In our recent paper "Prime numbers and random walks in a square grid" published in Phys. Rev. E
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.104.054114
we conjectured that the area of the Prime Random Walk keeps growing with the number of steps indefinitely.
Can you prove or disprove it?
We (A. Fraile, D. Fernández) offer 1000 $ to the person who proves it right, or wrong.
* The proof has to be published in a peer reviewed well respected international journal to ensure its validity.
** Offer expires in 2 years counting from the date of publication of this offer (18-4-2024), so, the 18-4-2026.
*** The proof must be complete, i.e. not based of some unproved conjectures or hypotheses.
Alberto Fraile
0000-0003-2646-5159
Daniel Fernández
0000-0002-6896-1006
Essentially this is a question about the proportion of primes up to x that lie in a given relatively prime congruence class. Dirichlet's Theorem tells you that, on average, these proportions are the same, so you will always return to, approximately, your starting region. However, this does not rule out one congruence class taking the lead by a significant amount. If you assume the Generalised Riemann Hypothesis, then any congruence class will be in the lead for a positive proportion of x. However, we need to be able to quantify this to answer whether your area is bounded or not.
Let me answer a slightly different question. You are moving on a line and move one step to the right if the prime is 4k+1 and to the left if it is 4k+3. A theorem of Littlewood from 1914 shows that in this case you will be able to move arbitrarily far from the origin.
I'm not aware of such a precise result for larger moduli (I'm not a number theorist!), but probably the answer to your question is that the area is unbounded, but proving this (and without the Generalised Riemann Hypothesis) will likely take much more than two years. (Bear in mind as well that the review process to get such an article published could also take more than two years in itself).
Have a look at the article 'Prime Number Races' by Granville and Martin. By the way, I suggest putting up a link to the arXiv version of your paper, as this version is behind a paywall :)