Number theory and Group theory

Using the definitions of p-adic convergence we can compute very interesting convergences regarding the factorial function. Claim. å_n=0^¥ nn! = -1. Proof: We say that a = å n!. We do not know if a is rational or not, but we can compute the following: å_n=0^¥ ((n+1)! - n!) = 1! -0! + 2! - 1! + 3! -2! + ... = -1. Because of the alternating sign, when we go to infinity all the terms will cancel except for -0!, which is equal to -1. Then (n+1)! - n! = n!(n+1-1) = nn! å nn! = å nn! = -1. The following output of our C++ code illustrates the meaning of p-adic convergence to -1 in Z₅. Note that the digit p-1, in this case 4 because -1 º 4 modulo 5 keeps repeating in the end more times as n increases.

Then we can compute å n² n! using the following trick: å_n=0 (n+2)(n+1)n! = å_n=0 (n+2)! = å_n=2 n! = a - 1. However, we also have that å_n=0 (n+2)(n+1)n! = å_n=0 (n² + 3n + 2)n! = å_n=0 n² n! + 3 å_n=0 nn! + 2 å_n=0 n!. Then substitutin: å_n=0 n² n! + 3 å_n=0 nn! + 2 å_n=0 n! = å_n=0 n² n! -3 + 2(a +1). Finally equating: a -1 = å_n=0^¥ n² n! -1 + 2a å_n=0 n² n! = -a. This is a recursive process and by obtaining all the values of te convergence of å_n=0 n^k-1 n! we can obtain the value of å_n=0 n^k n!. We also note the following: Claim. If å_n=0^¥ n! is rational, then any å_n=0^¥ n^k n! is also rational. Proof: As it is seen from the recursion, it is clear that å_n=0^¥ n^k n! = a a + b, where the values of a and b depend on k. Therefore, if a is rational, then so is a a + b. Following the recursivity, we can compute the first values of å_n=0 n^k n!: å_n=0 n³ n! = a + 2 å_n=0 n⁴ n! = 2 a -3 å_n=0 n^k n! = -9 a - 4. It is interesting to observe that if the coefficient of a is 0, then å_n=0 n^k n! converges to an integer number. We have computed using C++ the coefficients of a until very large numbers, and we have encountered that it only seems to be 0 for k=1. Table 3 shows the values of the coefficient of a until k=15. We observe that the value of the coefficient of a bounces between positive and negative values, but the absolute value seems to keep increasing. This provides evidence to conjecture the following: Conjecture 1. The series ån=0 n^k n! converge to an integer number only when k=1.

We can also generatlize these series and study the convergence of å_n=1 n^k (n+m)!. Using a similar recursive method, we wrote a C++ that computed the coefficient of alpha of these series for each value of k and m. We found the following: Conjecture. The series å_n=1 n^k (n+m)! only converge to an integer for k=1, m=0,k=2, m=1 and k=5, m=1.