1 Natural numbers

Definition 1 Natural numbers

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1.1 Addition

Here we define addition of natural numbers.

Definition 2

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Natural number addition.

Theorem 3

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For any natural number \(a\), \(0 + a = a\), where \(+\) is Def. 2.

Proof ▶

The proof follows by induction.

Theorem 4

For any natural numbers \(a, b\), \((a + 1) + b = (a + b) + 1\).

Proof ▶

Proof by induction on b.

Theorem 5

For any natural numbers \(a, b\), \(a + b = b + a\).

Proof ▶

The base case follows from 3.

The inductive case follows from 4.

1.2 Multiplication

Definition 6

Natural number multiplication.

Theorem 7

For any natural numbers \(a, b\), \(a * b = b * a\).

Proof ▶

1.3 Fermat’s Last Theorem

Theorem 8 Taylor–Wiles

# Discussion

Fermat’s last theorem.

Proof ▶

See [ 1 , 2 ] .

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1: Andrew Wiles (1995) Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics, 141(3), 443–551.
2: Richard Taylor and Andrew Wiles (1995) Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics, 141(3), 553–572.