In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture.

The aim of the present paper is to suggest that statistical physics provides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory.

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.