Problem 3 - A Gentle Introduction to the Art of Mathematics

Question

The alternating sum of factorials provides an interesting example of a sequence of integers. $$ \begin{array}{c} 1 !=1 \\ 2 !-1 !=1 \\ 3 !-2 !+1 !=5 \\ 4 !-3 !+2 !-1 !=19 \end{array} $$ et cetera Are they all prime? (After the first two 1 's.)

Step-by-Step Solution

Answer

The alternating sum of factorials remains prime for n=3, 4, 5, and 6 as observed.

Step 1: Identify the Pattern

First, observe the given sequence. The general form of the alternating sum of factorials can be expressed as: \[ a_n = n! - (n-1)! + (n-2)! - (n-3)! + \ldots \pm 1! \]

Step 2: Base Cases

Confirm the given values are correct for n = 1 to 4: 1! = 1 2! - 1! = 1 3! - 2! + 1! = 6 - 2 + 1 = 5 4! - 3! + 2! - 1! = 24 - 6 + 2 - 1 = 19

Step 3: Calculate Additional Terms

Check if the next numbers in the sequence are prime. Start with n = 5: \[ 5! - 4! + 3! - 2! + 1! = 120 - 24 + 6 - 2 + 1 = 101 \] 101 is a prime number. Next, for n = 6: \[ 6! - 5! + 4! - 3! + 2! - 1! = 720 - 120 + 24 - 6 + 2 - 1 = 619 \] Check if 619 is prime.

Step 4: Check Primality of 619

To verify if 619 is prime, check for divisibility by any prime numbers less than its square root (approximately 24.9): 619 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23. Therefore, 619 is a prime number.

Step 5: General Observation

Based on the findings so far, the alternating sum of factorials results in prime numbers for n = 3, 4, 5, and 6. To make a stronger generalization, further calculations for larger n-values would be necessary. Additionally, it's crucial to point out that proving a pattern for all integers n typically requires a different approach other than computation.

Key Concepts

Factorials

Factorials are a foundational concept in mathematics, especially in combinatorics. The factorial of a positive integer n, denoted as \(n!\), is the product of all positive integers less than or equal to n. For instance:

• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorials grow extremely fast. By the time you reach \(10!\), you're already at 3,628,800.

Factorials have many applications, such as in calculating permutations and combinations. They are also integral to understanding the sequence given in the exercise. For example, in the sequence, 4! or 24 is part of the alternating sum.

Prime Numbers

Prime numbers are another essential concept in mathematics. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Some examples include:

• 2
• 3
• 5
• 7

Checking if a number is prime becomes more complex as numbers get larger.

To verify if a number like 619 is prime, we check its divisibility by all prime numbers less than its square root. For 619, this means checking for divisibility by 2, 3, 5, 7, 11, 13, 17, 19, and 23. Since none of these primes divide 619, it is also a prime number. This is critical when analyzing sequences like the alternating sum of factorials to see if they consistently produce prime numbers.

Alternating Sums

An alternating sum is a sequence where terms are alternately added and subtracted. An example is the given exercise's factorials sequence:

\[ a_n = n! - (n-1)! + (n-2)! - (n-3)! + \ \ \ldots \ \ \pm 1! \]

For n=4, this becomes:

\ \ 4! - 3! + 2! - 1! = 24 - 6 + 2 - 1 = 19 \ \ Alternating sums are used in many areas of mathematics and help balance out sequences to create unique patterns, just like finding primes in this exercise.

Alternate sums add complexity to a sequence but can sometimes show otherwise hidden patterns, like the recurring prime numbers seen here.

• They provide fascinating insights when combined with other mathematical concepts.