Normal numbers and the multiplicative structure of integers

正规数与整数的乘法结构 正规数是一个无理数，它的性质是，从0到9的每个数字平均以预期的频率(即1/10)出现在其小数展开中，任何给定的一对数字(比方说37)也以1/100的频率出现在其小数展开中，依此类推。人们认为数字=3.141592是一个正态数字，尽管从来没有人能够证明这一点。同样，常见的常量，如SQRT 2和LOG 2，从未被证明是正态数，尽管数字证据有力地表明它们是正态数。事实上，证明一个给定的数字是正常的是一项非常困难的任务。然而，我们已经成功地构造了各种正规数族。例如，设P(N)表示整数n&gt；1的最大素因数，我们证明了实数0.P(2)P(3)P(4)P(5)P(6)=0.23253是正态数。数学界对正态数很感兴趣，特别是因为正态数可以用来产生所谓的伪随机数。我们正在研究的另一个问题与整数的乘法结构有关，即将整数表示为素数的乘积。事实上，众所周知，每个整数n&gt；1都可以写成素数的乘积。例如，60=22x3x5。我们感兴趣的是寻找连续的整数，每个整数都可以被其最大素因数的幂整除。当然，找到两个连续的整数，每个整数都可以被它们的最大素因数的平方整除是一项容易的任务，因为例如，49=72和50=2 x 52，实际上人们可以证明存在无限多对这样的整数。三个连续的整数n，n1，n2，每个都可以被其最大素因数的平方整除，情况如何？使用计算机，人们很快就会发现数字n=1294298具有这一性质。四个连续的整数，每个可被其最大素因数的平方整除，情况如何？没有找到这样的数字，尽管启发式的论点表明，这样的四倍体是存在的，而且事实上，它们的数量是无限的。更广泛地说，我们能找到k个连续的整数，每个都可以被它们的最大素因数的r次方整除吗？对于k=3和r=2，我们找到了许多解，但当k=3时，我们遇到了障碍。为了解决这样的问题，我们决定着眼于更大的图景，即通过在x中创建连续的多项式，每个多项式都有一个平方因子，目标是通过用正确的值替换x，我们将发现每个连续的整数都可以被其最大素因数的幂整除。而且，在大多数情况下，它是有效的！这个问题对数学家非常感兴趣，因为解决它，即使是部分地，也会揭示整数的乘法结构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.