Big Maths Ideas · Beta

Fermat’s Last Theorem

Compare squares and cubes, search integer pairs, and test a misleading near miss.

32+42=523^{2}+4^{2}=5^{2}
a = 3
a2=9a^{2}=9
b = 4
b2=16b^{2}=16
c = 5
c2=25c^{2}=25

Square areas are shown on a common length scale.

Exact match. This square example is a Pythagorean triple.

aⁿ + bⁿ
25
cⁿ
25
Left − right
0
Required positive real root
5
Exact root check
a2+b22=5\sqrt[2]{a^{2}+b^{2}}=5

Try another Pythagorean triple, then change n to 3. Fermat’s theorem starts above exponent 2.

The theorem and this experiment

For every integer exponent n greater than 2, no positive integers a, b and c satisfy an+bn=cna^n+b^n=c^n. Squares, where n = 2, are outside the theorem.

A positive real root always exists. Fermat’s theorem concerns positive integers. A finite search and a rounded display cannot prove the theorem.

Andrew Wiles’s proof uses advanced mathematics. Read the original paper’s journal record.

Predict, test and explain

Step 1 of 3

Predict

For a = 3, b = 4 and c = 5, does the equality still hold when squares become cubes?

InvestigateWhy does 3² + 4² = 5² work, while the same lengths fail for cubes?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Does 2³ + 3³ equal the cube of a positive integer? Does checking this one pair prove Fermat’s Last Theorem?

23+33=c3  ?2^3+3^3=c^3\;?

Hint

Calculate the sum, then compare the consecutive cubes on either side.

Reveal answer

The sum is 35. It lies strictly between 3³ = 27 and 4³ = 64, so its cube root is not an integer. This rules out this pair only; it is not a proof for every positive integer.

27<35<6427<35<64

Build an explanation

Hint 1

Start with n = 2 and a = 3, b = 4, c = 5. Compare the two smaller square areas with the larger one.

Hint 2

Keep the integers fixed and switch to n = 3. Calculate both sides exactly; a whole-looking decimal root is not enough.

Hint 3

In the search, an integer root would satisfy cⁿ = aⁿ + bⁿ exactly. The search covers only its stated finite range, not every positive integer.

Worked example

This example uses fixed values, separate from the diagram controls.

Does 3³ + 4³ equal the cube of a positive integer?

  1. Calculate exactly: 3³ + 4³ = 27 + 64 = 91.
  2. Compare consecutive cubes: 4³ = 64 and 5³ = 125.
  3. Since 64 < 91 < 125, its positive cube root lies strictly between 4 and 5.
  4. This proves that this particular sum is not an integer cube. It does not prove Fermat’s theorem for every integer.

Answer: No. The exact sum 91 lies between the consecutive cubes 64 and 125.

Connect this idea

Watch out: Fermat’s Last Theorem concerns positive integers and integer exponents greater than 2. A noninteger real root is allowed. An empty finite search or a rounded calculator display is not a proof of the theorem.