Fermat’s Last Theorem: The Most Famous Problem in Number Theory
– Pierre de Fermat, around 1637
The Statement
For any integer n > 2, there are no positive integers x, y, and z that satisfy:
xⁿ + yⁿ = zⁿ
When n = 2 (Pythagorean triples exist):
3² + 4² = 5²9 + 16 = 25 ✓
But for n = 3:
3³ + 4³ ≠ 5³27 + 64 = 91 ≠ 125
And for n = 4:
2⁴ + 3⁴ ≠ 4⁴16 + 81 = 97 ≠ 256
This Week’s Problems
1. Understanding n = 2
Find three different sets of positive integers (x, y, z) that satisfy:
x² + y² = z²
Hint: Look for numbers smaller than 20.
2. Testing n = 3
- Calculate 2³ + 2³ and compare it to 3³
- Calculate 3³ + 3³ and compare it to 4³
- What do you notice about these differences?
3. Pattern Investigation
For n = 2, we know 3² + 4² = 5²
Try to raise all numbers by the same power:
- Calculate: 3³ + 4³ compared to 5³
- Calculate: 3⁴ + 4⁴ compared to 5⁴
- What happens to the difference as n increases?
Historical Context
This seemingly simple statement remained unproven for 358 years until Andrew Wiles published his proof in 1995. The proof uses sophisticated mathematics developed in the 20th century, including:
- Modular Forms
- Elliptic Curves
- Galois Representations
Consider the contrast:
When n = 2: Many solutions exist (Pythagorean triples)
Examples:
3² + 4² = 5²5² + 12² = 13²
8² + 15² = 17²
When n > 2: No solutions exist (Fermat’s Last Theorem)
Challenge Questions
1. Without calculating, explain why 2ⁿ + 2ⁿ can never equal 3ⁿ for any n ≥ 2.
2. Show that if Fermat’s Last Theorem is true for n, it’s also true for all multiples of n.
Solution
The Explosive Growth in Fermat’s Last Theorem
Why does xⁿ + yⁿ = zⁿ have no solutions when n > 2? Let’s visualize the growth!
Starting with Our Known Solution
We know 3² + 4² = 5²
Let’s see what happens when we increase the power…
| Power | 3ⁿ | 4ⁿ | Sum | 5ⁿ | Gap |
|---|---|---|---|---|---|
| n = 2 | 9 | 16 | 25 | 25 | 0 |
| n = 3 | 27 | 64 | 91 | 125 | 34 |
| n = 4 | 81 | 256 | 337 | 625 | 288 |
| n = 5 | 243 | 1,024 | 1,267 | 3,125 | 1,858 |
Key Observation
Look at the “Gap” column – it shows how much bigger 5ⁿ is than (3ⁿ + 4ⁿ).
This gap grows explosively: 0 → 34 → 288 → 1,858
Why Does This Happen?
As we increase the power:
- Larger numbers grow faster than smaller numbers
- 5ⁿ grows much faster than 3ⁿ + 4ⁿ
- Once a gap appears (at n = 3), it becomes impossible to close
Exponential Growth Visualization
Think about doubling each time:
- 2¹ = 2
- 2² = 4 (doubled)
- 2³ = 8 (doubled again)
- 2⁴ = 16 (doubled again)
Now imagine this pattern with larger numbers like 3, 4, and 5!
The Big Picture
This is why Fermat’s Last Theorem works:
- For n = 2: Perfect balance is possible (Pythagorean triples)
- For n > 2: The larger number grows too fast to ever be caught by the sum of the smaller numbers
