Normal numbers and the multiplicative structure of integers

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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 p=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.