Natural numbers, their successor structure, and the basic operations (addition, multiplication) defined recursively on them. Built on the Peano axioms.
arithmetic
Zero is a natural number
0 is a natural number.
Every natural number has a successor
For every natural number n, S(n) is also a natural number.
Principle of mathematical induction
If a property holds for 0 and is preserved under succession, then it holds for all natural numbers.
Natural numbers (ℕ)
The set {0, 1, 2, 3, ...} constructed via Peano axioms.
Addition on naturals
Binary operation defined recursively: a+0 = a; a+S(b) = S(a+b).
Multiplication on naturals
Binary operation defined recursively: a·0 = 0; a·S(b) = a·b + a.
Peano arithmetic (PA)
First-order theory of arithmetic axiomatised by Peano's axioms (0 is a natural, successor is injective and avoids 0, plus the induction…
Robinson arithmetic (Q)
A finitely axiomatised, quantifier-free fragment of PA. Already strong enough to support Gödel's first incompleteness theorem despite…
Primitive recursive function
The smallest class of functions ℕ^k → ℕ containing zero, successor and projections and closed under composition and primitive recursion. …
Presburger arithmetic
First-order theory of ⟨ℕ, 0, 1, +, <⟩ (no multiplication). Presburger (1929) proved it is decidable, complete, and admits quantifier…
Gentzen's consistency proof for PA
Gentzen (1936): PA is consistent, proved by transfinite induction up to the ordinal ε₀ — outside PA by the second incompleteness theorem. …
Heyting arithmetic (HA)
The intuitionistic counterpart of PA: same axioms, but built over intuitionistic predicate logic. HA ⊢ φ ⇒ PA ⊢ φ but the converse fails…
Ostrowski's theorem
Every non-trivial absolute value on ℚ is equivalent to either |·|_∞ or a p-adic |·|_p. Classification of completions of ℚ.
Hensel's lemma
If f(x) ∈ ℤ_p[x], a₀ ∈ ℤ_p with |f(a₀)| < |f'(a₀)|², then unique a ∈ ℤ_p with f(a)=0, a ≡ a₀ mod p. Newton's method over p-adics.
Local-global (Hasse-Minkowski)
Quadratic form over ℚ represents 0 nontrivially iff it does over every completion ℝ and ℚ_p. Fails for higher degree (Selmer…
Chebotarev density theorem
In Galois extension L/K with group G, Frobenius conjugacy classes are equidistributed: density of primes with Frob_p in conjugacy class C…
Mordell-Weil theorem
For elliptic curve E/K (number field): E(K) is a finitely-generated abelian group. E(K) ≅ ℤ^r ⊕ T with r = rank, T torsion.
Modularity theorem (Taniyama-Shimura-Weil)
Every elliptic curve over ℚ is modular: L(E,s) = L(f,s) for weight-2 newform f. Wiles-Taylor 1995 (FLT); full BCDT 2001.
Fermat's Last Theorem
No nontrivial integer solutions to xⁿ + yⁿ = zⁿ for n ≥ 3. Wiles 1995 via modularity of semistable elliptic curves + Frey-Ribet.
Iwasawa theory
Studies growth of class numbers + Selmer groups in ℤ_p-extensions. Main conjecture: characteristic ideal equals p-adic L-function.…
Selmer / Tate-Shafarevich
0 → E(K)/nE(K) → Sel_n(E/K) → Ш(E/K)[n] → 0. Selmer is computable, Sha measures obstruction to Hasse principle on E.
ℓ-adic Tate module
T_ℓ(E) = lim← E[ℓⁿ], free ℤ_ℓ-module of rank 2 with Gal action. Connects geometry to Galois reps; foundation of étale cohomology.
Faltings' theorem (Mordell)
Smooth projective curve C/ℚ of genus ≥ 2 has finitely many rational points. Faltings 1983 (Fields Medal).
abc conjecture (open)
For coprime a+b=c with ε>0: c ≤ K_ε · rad(abc)^{1+ε} except finitely many triples. Mochizuki's IUT proof disputed.
Functional equation ξ(s)=ξ(1−s)
ξ(s) = ½s(s−1)π^{−s/2}Γ(s/2)ζ(s) satisfies ξ(s)=ξ(1−s). Analytic continuation of ζ; symmetry about s=½.
Dirichlet/Dedekind L-functions
L(s,χ) = Σ χ(n) n^{-s} for Dirichlet character; Dedekind ζ_K(s) = Σ_a N(a)^{-s}. Encodes class number formula: res_{s=1} ζ_K(s) =…
Analytic class number formula
res_{s=1} ζ_K(s) = (2^{r_1}(2π)^{r_2} · h_K · R_K)/(w_K · |d_K|^{1/2}). Class number, regulator, unit root, discriminant.
Cyclotomic field ℚ(ζₙ)
Splitting field of xⁿ−1. Gal(ℚ(ζₙ)/ℚ) ≅ (ℤ/n)*. Ring of integers ℤ[ζₙ]; class number h_n controls regular primes.
Kronecker-Weber theorem
Every finite abelian extension of ℚ is contained in a cyclotomic field ℚ(ζₙ). Abelian CFT over ℚ; generalization to CM fields (Kronecker…
Artin reciprocity
For abelian extension L/K: Frobenius element at unramified prime p depends only on p mod conductor. Generalises quadratic reciprocity; core…
Roth's theorem
Algebraic α of degree ≥ 2: |α − p/q| > C/q^{2+ε} for all p/q, finitely many exceptions. Thue-Siegel-Roth 1955 (Fields).
Weil height / Néron-Tate height
Logarithmic projective height h(P) = log max|x_i|; canonical ĥ on E quadratic form detecting rank. Key tool in Diophantine geometry.
Euler product formula
For Re(s)>1: ζ(s) = Π_p (1 − p^{-s})^{-1}. Generalises to L-functions; encodes unique factorisation.
Wilson's theorem
A natural number p > 1 is prime if and only if (p − 1)! ≡ −1 (mod p). Stated by Wilson 1770, first proved by Lagrange 1771. Proof: pair…
Sum-of-two-squares theorem
A positive integer n is the sum of two integer squares iff every prime p ≡ 3 (mod 4) occurs to an even power in its factorisation. Fermat…
Lagrange's four-square theorem
Every non-negative integer is a sum of four integer squares. Lagrange 1770 proof uses Euler's four-square identity (the product of two…
Möbius inversion
For arithmetic functions: if g(n) = Σ_{d|n} f(d) then f(n) = Σ_{d|n} μ(n/d) g(d), where μ is the Möbius function (μ(1)=1, μ(squarefree with…
Legendre's formula (factorial valuation)
The p-adic valuation of n! equals Σ_{k ≥ 1} ⌊n/p^k⌋. Equivalent form: v_p(n!) = (n − s_p(n))/(p − 1), where s_p(n) is the sum of base-p…
Stirling's approximation for n!
As n → ∞, n! ≈ √(2πn) · (n/e)^n with relative error O(1/n). Proved via Euler-Maclaurin, complex Laplace method, or the Γ-function integral.…
Bertrand's postulate
For every integer n ≥ 1, there is at least one prime in the interval (n, 2n]. Conjectured by Bertrand 1845, proved by Chebyshev 1852,…
Abel summation by parts
Discrete integration-by-parts: Σ a_n b_n = A_N b_N − Σ A_n (b_{n+1} − b_n), A_n the partial sums of a. Abel 1826. Tool for Dirichlet…