Introduction

The problem at hand explores the question of how many possible values of n make the product of factorials from 1! to n! a perfect square. This intriguing problem combines elements of number theory and factorial growth, offering valuable insights into the behavior of sequences and their square properties.

Understanding the Factorial Sequence

The factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For instance, 5! 5 × 4 × 3 × 2 × 1 120. Understanding the properties of this sequence is crucial to determining when a product of these factorials is a perfect square.

Initial Observations and Calculations

Let's start by calculating the value of k 1! 2! 3! ? n! for small values of n to observe any patterns or properties.

For n 1:
k 1! 1
1 is a perfect square. For n 2:
k 1! 2! 1 2 3
3 is not a perfect square. For n 3:
k 1! 2! 3! 1 2 6 9
9 is a perfect square. For n 4:
k 1! 2! 3! 4! 1 2 6 24 33
33 is not a perfect square. For n 5:
k 1! 2! 3! 4! 5! 1 2 6 24 120 153
153 is not a perfect square. For n 6:
k 1! 2! 3! 4! 5! 6! 1 2 6 24 120 720 873
873 is not a perfect square. For n 7:
k 1! 2! 3! 4! 5! 6! 7! 873 5040 5913
5913 is not a perfect square. For n 8:
k 1! 2! 3! 4! 5! 6! 7! 8! 5913 40320 46233
46233 is not a perfect square. For n 9:
k 1! 2! 3! 4! 5! 6! 7! 8! 9! 46233 362880 409113
409113 is not a perfect square. For n 10:
k 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 409113 3628800 4037913
4037913 is not a perfect square.

From these calculations, it is clear that k is a perfect square for n 1 and n 3, but for n ≥ 4, the product of factorials grows too rapidly to remain a perfect square.

Extending the Analysis Using Modulo Arithmetic

Further exploration into the problem reveals that another method involving modulo arithmetic can be used to determine whether a product of factorials is a perfect square. Specifically, using modulo 19, we observe that:

For n ≥ 18:

1! 2! 3! ? n! ≡ 8 (mod 19)

Since 8 is not a quadratic residue modulo 19, it cannot be a perfect square. Therefore, we only need to check for values of n ≤ 17.

Using brute force, we can confirm that n 1 and n 3 are the only values that make the product 1! 2! 3! ? n! a perfect square.

Conclusion

In conclusion, the only values of n for which the product of factorials from 1! to n! is a perfect square are n 1 and n 3.