number-theory

An integer p > 1 whose only positive divisors are 1 and p. The multiplicative atoms of the integers.

Every integer n > 1 has a unique factorisation into prime powers (up to order). Proven from strong induction on divisibility.

There are infinitely many primes. Euclid's proof: any finite list {p₁…pₙ} admits a prime factor of p₁·…·pₙ + 1 outside the list.

gcd(a,b) computed by repeated division: gcd(a,b) = gcd(b, a mod b), terminating at gcd(x,0) = x. Basis of Bezout's identity and of modular…

Arithmetic on integer residue classes modulo n: +, ·, and (when gcd(a,n)=1) multiplicative inverse. Backbone of cryptography (RSA, ECC) and…

For prime p and integer a with gcd(a,p)=1: a^(p-1) ≡ 1 (mod p). Generalised by Euler to a^φ(n) ≡ 1 (mod n). Basis of primality testing.

Number of integers in {1,…,n} coprime to n. Multiplicative, with φ(p) = p−1 for prime p. Generalises Fermat's little theorem.

If n₁,…,nₖ are pairwise coprime, ℤ/(n₁…nₖ) ≅ ℤ/n₁ × … × ℤ/nₖ as rings. Operational backbone of modular-arithmetic algorithms and of RSA-CRT.

All non-trivial zeros of ζ(s) = Σ 1/nˢ lie on the critical line Re(s) = 1/2. Unproven; Clay Millennium Problem. Implies sharp error bounds…

Every even integer n > 2 is the sum of two primes. Unproven; verified to > 4·10¹⁸ by computer search.

Density coefficient for integers expressible as a sum of two squares: #{n ≤ x : n = a²+b²} ~ K·x/√(ln x).

Mertens's theorem: Σ_{p≤n} 1/p = ln ln n + M + o(1). Characterizes the density of primes.

Conjectural density of primes for which a given non-square, non-(-1) integer is a primitive root. Unconditional proof depends on…

The smallest positive real A such that ⌊A^{3ⁿ}⌋ is prime for every n ≥ 1. Existence is proven unconditionally; the *exact* value below…

The unique positive root of the logarithmic integral li(x) = ∫₀ˣ dt/ln t. Used as the offset in the classical prime-counting estimate Li(x)…

The base-10 concatenation 0.123456789101112…; proven transcendental and a normal number in base 10.

The base-10 concatenation 0.235711131719…; built from prime digits. Proven normal in base 10 (Copeland & Erdős, 1946).

For almost every irrational x, the denominators q_n of the continued-fraction convergents grow like q_n^(1/n) → L. Closely tied to…

Sum of reciprocals of twin primes. Brun (1919) proved the sum converges — strong evidence that twin primes are much sparser than primes.…

Sum of reciprocals of prime quadruplets (p, p+2, p+6, p+8). Also converges (by Brun-type sieve arguments).

Hardy–Littlewood constant in the conjectural asymptotic π₂(x) ~ 2 C₂ x/(ln x)². Assumes the (unproven) twin-prime conjecture.

Asymptotic expected value of the logarithm of the largest prime factor of a random integer near n, divided by ln n. Appears in analysis of…

An alternating sum over Sylvester's sequence s_n (s_0=2, s_{n+1}=s_0·…·s_n + 1). Cahen (1891) proved C is transcendental.

Mean value of H(n) (the maximum exponent in the prime factorisation of a random integer). The value does NOT require the Riemann hypothesis…

Limit of ratios of successive coefficients in a power-series whose coefficients are built from primes.

The sum of the reciprocals of those positive integers whose decimal expansion contains no '9'. Kempner (1914) proved the series converges…

π(x) ~ x/ln x as x→∞ (Hadamard, de la Vallée Poussin 1896). Equivalent to ζ(s) having no zeros on ℜs=1; the error term sharpens under RH…

For any coprime integers a, q there are infinitely many primes ≡ a (mod q). Dirichlet (1837) introduced L-functions to prove it.

For distinct odd primes p, q: (p/q)(q/p) = (−1)^{(p−1)(q−1)/4}. Gauss's 'theorema aureum'; foundation of class field theory.

L(s,χ) = Σ χ(n)/n^s for a Dirichlet character χ mod q. Analytic continuation and non-vanishing on ℜs=1 yields Dirichlet's theorem; entire…

A polynomial equation f(x_1,…,x_n)=0 with integer coefficients sought over ℤ (or ℚ). Hilbert's 10th problem (Matiyasevich 1970): no…

For n ≥ 3 there are no positive integer solutions to x^n + y^n = z^n. Conjectured 1637, proved by Wiles (1995) via modularity of…

Completion of ℚ under the p-adic absolute value |x|_p = p^{−v_p(x)}. Complete, locally-compact, totally-disconnected field; together with…

A non-discrete locally-compact field: ℝ, ℂ, finite extensions of ℚ_p, or Laurent series 𝔽_q((t)). Classified by the Ostrowski/Pontryagin…

Restricted direct product Π'_v K_v of all completions of a global field K; admits a discrete embedding of K with compact quotient. Natural…

A smooth projective curve of genus 1 with a rational base point; equivalently y² = x³ + ax + b with 4a³+27b² ≠ 0. The points form an…

A holomorphic function f on the upper half-plane satisfying f((az+b)/(cz+d)) = (cz+d)^k f(z) for all (a,b;c,d) ∈ SL₂(ℤ) (or a congruence…

Describes abelian extensions of a (local or global) field K in terms of the multiplicative group K* / norms (local) or the idele class…

A continuous homomorphism ρ : Gal(\bar K/K) → GL_n(R) for a topological ring R (ℓ-adic, mod-p, etc.). Attaches linear data to arithmetic…

For an elliptic curve E/ℚ, the rank of E(ℚ) equals the order of vanishing of L(E,s) at s=1, and the leading Taylor coefficient has an…

A web of conjectures unifying number theory and representation theory: automorphic representations of reductive groups should correspond to…

For any integers a, b (not both zero), there exist integers x, y such that ax + by = gcd(a, b). The pair (x, y) is not unique; the…

Three asymptotic estimates on primes due to Mertens (1874): (1) Σ_{p ≤ n} (ln p)/p = ln n + O(1); (2) Σ_{p ≤ n} 1/p = ln ln n + M + o(1),…

gcd(a,d)=1 ⟹ infinitely many primes p ≡ a (mod d). Proof via L(s,χ) non-vanishing at s=1. Dirichlet 1837.

Primes contain arbitrarily long arithmetic progressions. Szemerédi + Gowers uniformity + Goldston-Yildirim. Green-Tao 2008.

lim inf (p_{n+1} − p_n) < ∞. Zhang 2013 (≤70M), Maynard 2013 (≤600), Polymath 8 (246). Twin-prime conjecture (∞-many with gap 2) open.

ζ(s) meromorphic on ℂ with simple pole at s=1 (residue 1). Via Γ-function contour, Mellin transform of θ, or Euler-Maclaurin.

ζ(1+it) ≠ 0 for real t, equivalent to PNT. Refinements: zero-free regions of Vinogradov, Korobov; feeds into best PNT error term.

g(k): smallest s such that every n = x_1^k + … + x_s^k. Hilbert 1909 existence; Hardy-Littlewood circle method; G(k) < g(k) uses fewer…

Additive problems via f(α) = Σ e(α·n): major arcs (near rationals) give main term, minor arcs controlled by Weyl / Vinogradov bounds.

Estimates cardinality of sets avoiding prescribed congruence conditions. Brun, Selberg, large sieve, Bombieri-Vinogradov. Base for gap…

g(χ) = Σ χ(n)ζ^n for Dirichlet χ mod q. |g(χ)| = √q (primitive); factorization of Gauss sums in ℤ[ζ_q] via Stickelberger element.

Compactified quotient ℍ/Γ_0(N); moduli space of elliptic curves with Γ_0(N)-structure. Hecke operators, newforms (Atkin-Lehner).

Commuting self-adjoint operators on space of modular forms. Simultaneous eigenforms yield Euler products L(f,s) = Π (1 − a_p p^{-s} +…

Hardy-Ramanujan asymptotic formula for the unrestricted partition function: p(n) ~ exp(pi*sqrt(2n/3)) / (4 n sqrt 3) as n -> infinity.…