Investigate special prime number formations involving exponential expressions and investigate if these primes have a specific mathematical name | Step-by-Step Solution

What We're Solving

We're investigating two specific types of expressions to see which ones produce prime numbers:

• 1. Numbers of the form 10^x - 10^y - 1 (where x > y)
• 2. Numbers of the form (10^x - 1)/y

We want to find examples and see if these have special names in mathematics!

The Approach

We'll systematically test small values to find patterns, then research whether mathematicians have already studied these forms. This teaches us how mathematicians explore number patterns and make discoveries.

Why this matters: Many famous mathematical discoveries started exactly this way - by noticing patterns in specific number forms.

Step-by-Step Solution

Part 1: Exploring 10^x - 10^y - 1

Step 1: Start with small values where x > y ≥ 0

• Let's try x = 2, y = 0: 10² - 10⁰ - 1 = 100 - 1 - 1 = 98 = 2 × 49 (not prime)
• Try x = 2, y = 1: 10² - 10¹ - 1 = 100 - 10 - 1 = 89

Step 2: Check if 89 is prime 89 is not divisible by 2, 3, 5, or 7, and since √89 < 10, we only need to check up to 9. Yes, 89 is prime! ✨

Step 3: Continue systematically

• x = 3, y = 0: 10³ - 10⁰ - 1 = 999 = 3³ × 37 (not prime)
• x = 3, y = 1: 10³ - 10¹ - 1 = 989 = 23 × 43 (not prime)
• x = 3, y = 2: 10³ - 10² - 1 = 899 = 29 × 31 (not prime)

Part 2: Exploring (10^x - 1)/y

Step 1: This form creates interesting patterns!

• x = 2, y = 9: (10² - 1)/9 = 99/9 = 11 (prime!)
• x = 3, y = 9: (10³ - 1)/9 = 999/9 = 111 = 3 × 37 (not prime)

Step 2: Notice the pattern - we're getting repunits! When y = 9, we get numbers like 11, 111, 1111... These are called repunits (repeated units).

Step 3: Try other divisors

• x = 2, y = 3: (10² - 1)/3 = 99/3 = 33 = 3 × 11 (not prime)

Part 3: Research Connection

The form (10^x - 1)/9 gives us repunits, and repunit primes are well-studied! The first few repunit primes are:

• R₂ = 11
• R₁₉ = 1111111111111111111 (19 ones)
• R₂₃ = 11111111111111111111111 (23 ones)

The Answer

Primes found:

• 1. Form 10^x - 10^y - 1: We found 89 (when x=2, y=1)
• 2. Form (10^x - 1)/y: We found 11 (when x=2, y=9)

Special names:

• The second form, when y=9, produces repunit primes - a famous class of primes consisting entirely of 1's
• The first form doesn't have a standard name but represents a novel pattern worth further investigation

Your research mission: Try more values! Can you find other primes of the form 10^x - 10^y - 1? This could be original mathematical research!

Memory Tip

Remember "Repunit = Repeated Units" - they're primes made entirely of 1's, like 11, 1111111111111111111. They come from the pattern (10^n - 1)/9, which creates strings of 1's!

Great work exploring these patterns - you're thinking like a real number theorist! 🌟