Finding the Limit of the Largest Power of 5 Dividing a Generalized Product
In this article, we delve into a fascinating problem in number theory involving the largest power of 5 dividing a particular product. Specifically, we investigate the sequence En, where En is defined as the largest integer k such that 5^k divides the product 1^1 cdot 2^2 cdot ldots cdot n^n.
Definition and Initial Insights
The function En can be defined as follows:
Let
First, observe that the nonzero contributions to this function only occur from the multiples of 5 leqslant n. Each multiple of 5 contributes a value that is a sum of series, which can be represented as:
[sum_{k1}^{leftlfloor frac{n}{5} rightrfloor} 5k]
Further Contributions and Infinite Series
However, this is not the complete story. Additional contributions arise from multiples of etc. Each time a factor of is considered, we need to account for the additional factor of etc.
The formula for the total contribution from these additional factors is given by:
[sum_{ell1}^{lfloor log_5{n} rfloor} sum_{k1}^{leftlfloor frac{n}{5^{ell}} rightrfloor} 5^{ell}k]
This expression is finite because for sufficiently large and for all , the term (leftlfloor frac{n}{5^j} rightrfloor 0).
Limit Calculation
The central question is to find the limit of the expression (lim_{n to infty} frac{E_n}{n^2}). To approach this, we use the poverty growth estimates, noting that .
The problem can then be simplified to:
[lim_{n to infty} frac{1}{n^2} sum_{ell1}^{lfloor log_5{n} rfloor} sum_{k1}^{leftlfloor frac{n}{5^{ell}} rightrfloor} 5^{ell}k]
This further simplifies to:
[lim_{n to infty} cdots lim_{n to infty} frac{1}{2} sum_{ell1}^{lfloor log_5{n} rfloor} left(frac{1}{5^{ell}} text{O}left(frac{log_5{n}}{n}right)right)]
By letting n to infty, we observe that , which leaves us with a convergent geometric series:
[lim_{n to infty} frac{1}{2} sum_{ell1}^{infty} frac{1}{5^{ell}} frac{1}{2} cdot frac{frac{1}{5}}{1 - frac{1}{5}} boxed{frac{1}{8}}]
Generalization to Any Prime
The same argument can be generalized for any fixed prime p. If En denotes the largest integer k such that p^k divides the product 1^1 cdot 2^2 cdot ldots cdot n^n, then the limit is given by:
[lim_{n to infty} frac{E_n}{n^2} boxed{frac{1}{2p-1}}]
This result highlights the importance of understanding the distribution of prime powers in sequences like 1^1, 2^2, ldots, n^n and provides a powerful analytical tool for solving similar problems in number theory.