# Walking to Infinity Along Some Number Theory sequences

@article{Miller2020WalkingTI, title={Walking to Infinity Along Some Number Theory sequences}, author={Steven J. Miller and Fei Peng and Tudor Popescu and Joshua M. Siktar and Nawapan Wattanawanichkul and The Polymath Reu Program}, journal={arXiv: Number Theory}, year={2020} }

An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed… Expand

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SHOWING 1-8 OF 8 REFERENCES

The Difference Between Consecutive Primes, II

- Mathematics
- 2001

With enough effort, the value of x0 could be determined effectively. The paper has much in common with [1]; in particular we use the sieve method of Harman [4, 5]. We no longer use zero density… Expand

On the difference between consecutive primes

- Mathematics
- 2012

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new… Expand

Fibonacci and Lucas Numbers with Applications

- Mathematics
- 2001

Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes.… Expand

Stochastic Models for the 3x+1 and 5x+1 Problems

- Mathematics
- 2009

This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem. The stochastic models are… Expand

Beyond pair correlation

- Mathematics
- 2000

The authors study the distribution of psi(x+h)-psi(x)-h and compare it with numerical data.

Miller and R

- Takloo-Bighash,An Invitation to Modern Number Theory, Princeton University Press,
- 2006

Takloo-Bighash, An Invitation to Modern Number

- 2006

Beyond pair correlation, Paul Erdős and his mathematics

- Vol. I (Budapest,
- 1999