Normal numbers and the multiplicative structure of integers

正规数与整数的乘法结构 一个正常数是一个无理数，它具有这样的性质，即从0到9的每一个数字平均以预期的频率（即1/10）出现在它的十进制扩展中，任何给定的数字对，比如37，也以1/100的频率出现在它的十进制扩展中，等等。人们认为，这个数= 3.141592是一个正常数，类似地，常见的常数如sqrt 2和log 2从未被证明是正规数，尽管数值证据强烈地表明它们是。事实上，证明一个给定的数字是正常的是一个非常困难的任务。然而，我们已经成功地构造了各种各样的正规数族。例如，设P（n）表示n> 1的最大素因子，证明了真实的数0. P（2）P（3）P（4）P（5）P（6）= 0. 23253是正规数。在数学界中对正规数有很大的兴趣，特别是因为正规数可以用来生成所谓的伪随机数。我们正在研究的另一个问题与整数的乘法结构有关，即整数表示为素数的乘积。实际上，已知每个整数n> 1都可以写成素数的乘积。例如，60 = 22 x 3 x 5。我们感兴趣的是找到连续的整数，每个整数都能被它的最大素因子的幂整除。当然，找到两个连续的整数，每个整数都能被它们的最大素因子的平方整除是一件容易的事情，因为例如，49 = 72和50 = 2 x 52，事实上，人们可以证明有无限多对这样的整数。那么三个连续的整数n、n +1、n +2（每个都能被其最大素因子的平方整除）呢？ 使用计算机，人们会很快发现数n = 1294298具有这个性质。如果四个连续的整数都能被其最大素因子的平方整除，那又如何呢？没有发现这样的数字，尽管启发式的论点表明这样的四元组存在，事实上，有无限多个。更一般地说，我们能找到k个连续的整数，每个整数都能被它们的最大素因子的r次幂整除吗？对于k = 3和r = 2，我们找到了许多解决方案，但当k> 3时，我们遇到了障碍。为了解决这些问题，我们决定从更大的角度来考虑，即通过创建x中的连续多项式，每个多项式都有一个平方因子，目标是通过替换x的正确值，我们将发现连续整数，每个整数都可以被其最大素因子的幂整除。它的工作，大多数时候!这个问题对数学家来说非常有趣，因为即使是部分地解决它，也会揭示整数的乘法和加法结构之间的某种联系。

Normal numbers and the multiplicative structure of integers A normal number is an irrational number with the property that on average each of the digits from 0 to 9 occurs in its decimal expansion at the expected frequency, that is 1/10, that also any given pair of digits, say 37, occurs in its decimal expansion with a frequency of 1/100, and so on. It is believed that the number =3.141592 is a normal number, even though no one has ever been able to prove it. Similarly, common constants such as sqrt 2 and log 2 have never been proved to be normal numbers, even though numerical evidence strongly suggests that they are. In fact, proving that a given number is normal is a very difficult task. Nevertheless, we have been successful in constructing various families of normal numbers. For instance, letting P(n) stand for the largest prime factor of an integer n>1, we proved that the real number 0.P(2)P(3)P(4)P(5)P(6) = 0.23253 is a normal number. There is much interest for normal numbers in the mathematical community, in particular because normal numbers may be used to generate so-called pseudo random numbers. Another problem we are working on is related to the multiplicative structure of integers, that is, the representation of integers as a product of prime numbers. Indeed, it is known that each integer n>1 can be written as a product of prime numbers. For example, 60 =22 x 3 x 5. We are interested in finding consecutive integers each of which is divisible by a power of its largest prime factor. Of course, finding two consecutive integers each of which is divisible by the square of their largest prime factor is an easy task, because for instance, 49=72 and 50=2 x 52, and in fact one can prove that there are infinitely many such pairs of integers. What about three consecutive integers n, n+1, n+2, each divisible by the square of their largest prime factor? Using a computer, one will quickly find that the number n=1294298 has this property. What about four consecutive integers each divisible by the square of its largest prime factor? No such number has been found, although heuristic arguments indicate that such quadruples exist and, in fact, that there are infinitely many of them. More generally, can we find k consecutive integers each divisible by the r-th power of their largest prime factor? For k=3 and r=2, we found many solutions, but when k>3, we run into obstacles. To tackle such problems, we decided to look at the bigger picture, namely by creating consecutive polynomials in x each having a squared factor, the goal being that by substituting the correct value for x, we will discover consecutive integers each divisible by a power of their largest prime factor. And it works, most of the times! This problem is of great interest for mathematicians because solving it, even partially, will reveal some connection between the multiplicative and additive structures of integers.