The Prime Grid. Introducing a geometric representation of natural numbers

In this report we present an off-the-number-line representation of the positive integers by expressing each integer by its unique prime signature as a grid point of an infinite dimensional space indexed by the prime numbers, which we term the prime grid. In this space we consider a zigzag line, termed the number trail that starts at the origin (representing 1) and travels through every single grid point in the order of the increasing sequence of the natural numbers. Using the infinity norm we define an arithmetic function $L_\infty (N)$ tabulating the total length of the zigzag up to the integer $N$. We show that $L_\infty(N)$ grows linearly in $N$. Based on computing $L_\infty$ up to $N=10^{12}$ we conjecture the exact rate of growth, which we substantiate analytically by constructing a series of Markov shifts that give gradually better approximations. Our other interest is looking at the prime gaps along $L_\infty$, i.e. the differences $L_\infty (p_{i+1} )-L_\infty (p_i )$ between every two consecutive primes. After some preliminary observations we extend this analysis to second order differences (difference of differences) as well. Based on our computations up to $N=10^{12}$ we conclude that the distribution of prime numbers along the number trail shows a considerably richer structure compared to their distribution on the traditional number line. We also formulate modified versions of the prime number theorem and Polignac's conjecture.