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Showing 50 of 280 problems in Number theory

NT-AN-001

Riemann Hypothesis

IconicConjecture1859

Let

ζ (s) = n = 1 \sum \infty n^{- s} (ℜ s > 1)

and let the same symbol denote its meromorphic continuation to

C

, whose only pole is the simple pole at

s = 1

. Every zero

ρ

ζ

with

0 < ℜ ρ < 1

satisfies

ℜ ρ = \frac{1}{2}

NT-ARITH-002

Birch–Swinnerton-Dyer Conjecture

IconicConjecture1965

Let

E / Q

be an elliptic curve. Write

E (Q) ≅ E (Q)_{tors} \oplus Z^{r}

, and let

L (E, s)

be the Hasse–Weil

L

-function of

E

, continued to an entire function by modularity. Then

ord_{s = 1} L (E, s) = r .

NT-ADD-005

Strong Goldbach Conjecture

IconicConjecture1742

For every even integer

N \geq 4

, there exist prime numbers

p, q

(not necessarily distinct) such that

N = p + q

NT-PRIME-006

Twin Prime Conjecture

IconicConjecture1849

There are infinitely many primes

p

for which

p + 2

is also prime.

NT-DYN-014

Collatz Conjecture

IconicConjecturec. 1937

Define

T : Z_{> 0} \to Z_{> 0}

T (n) = {n /2, 3 n + 1, n even, n odd .

For every

n \in Z_{> 0}

, there exists

k \geq 0

such that

T^{k} (n) = 1

NT-AN-003

Generalized Riemann Hypothesis for Dirichlet $L$ -functions

LandmarkConjecturec. 1884

For every primitive Dirichlet character

χ

modulo an integer

q \geq 1

, every zero

ρ

of the analytically continued Dirichlet function

L (s, χ) = n = 1 \sum \infty \frac{χ ( n )}{n ^{s}} (ℜ s > 1)

satisfying

0 < ℜ ρ < 1

has

ℜ ρ = \frac{1}{2}

NT-PRIME-009

Bateman–Horn Conjecture

LandmarkConjecture1962

Let

f_{1}, \dots, f_{k} \in Z [x]

be distinct irreducible polynomials with positive leading coefficients. Assume their product

f = \prod_{i} f_{i}

has no fixed prime divisor, meaning that no prime divides

f (n)

for every

n \in Z

. For a prime

p

, set

N_{p} = # {a \in Z / p Z : f (a) = 0}

, and

C (f_{1}, \dots, f_{k}) = p \prod \frac{1 - N _{p} / p}{( 1 - 1/ p ) ^{k}} .

Then, as

x \to \infty

# {n \in Z : 1 \leq n \leq x, f_{1} (n), \dots, f_{k} (n) are all prime} \sim \frac{C ( f _{1} , \dots , f _{k} )}{\prod _{i = 1}^{k} de g f _{i}} \int_{2}^{x} \frac{d t}{( lo g t ) ^{k}} .

NT-TRANS-021

Schanuel’s Conjecture

LandmarkConjecturec. 1960

For every integer

n \geq 1

, if

z_{1}, \dots, z_{n} \in C

are linearly independent over

Q

, then

trdeg_{Q} Q (z_{1}, \dots, z_{n}, e^{z_{1}}, \dots, e^{z_{n}}) \geq n,

where

trdeg_{Q}

denotes transcendence degree over

Q

NT-LOGIC-025

Hilbert’s Tenth Problem over $\mathbb Q$

LandmarkOpen problem1970

There is no Turing machine which, on every input polynomial

f \in Z [x_{1}, \dots, x_{n}]

(where

n

is part of the input), halts and correctly decides whether there exists

(q_{1}, \dots, q_{n}) \in Q^{n}

with

f (q_{1}, \dots, q_{n}) = 0

NT-AUTO-030

Generalized Ramanujan Conjecture for $\mathrm{GL}_n$

LandmarkConjecturec. 1967

For every

n \geq 2

, every irreducible unitary cuspidal automorphic representation

π = ⨂_{v}^{'} π_{v}

GL_{n} (A_{Q})

, where

A_{Q}

is the adèle ring of

Q

, and every place

v

Q

, the local representation

π_{v}

is tempered; equivalently, all its

K_{v}

-finite matrix coefficients belong to

L^{2 + ε} (GL_{n} (Q_{v}) / Z_{v})

for every

ε > 0

, where

K_{v}

is a maximal compact subgroup and

Z_{v}

is the center.

NT-GALOIS-031

Fontaine–Mazur Conjecture

LandmarkConjecture1995

Let

p

be a prime, put

G_{Q} = Gal (\overline{Q} / Q)

, and let

ρ : G_{Q} ⟶ GL_{n} (\overline{Q}_{p})

be a continuous irreducible representation, unramified outside finitely many primes and potentially semistable at

p

, meaning semistable after restriction to the Galois group of a finite extension of

Q_{p}

. Then there exist a smooth projective variety

X / Q

, integers

i, j

, and a finite extension

E / Q_{p}

such that

ρ

, after scalar extension to

E

, is a subquotient of

H_{\overset{e}{ˊ} t}^{i} (X_{\overline{Q}}, E) (j)

NT-DIST-032

Elliott–Halberstam Conjecture

LandmarkConjecture1968–1970

Let

Λ (n) = lo g p

n = p^{m}

for a prime

p

and an integer

m \geq 1

, and let

Λ (n) = 0

otherwise. Put

ψ (y; q, a) = n \leq y n \equiv a (mod q) \sum Λ (n) and φ (q) = # (Z / q Z)^{\times} .

For every

0 < θ < 1

and every

A > 0

q \leq x^{θ} \sum 1 \leq a \leq q (a, q) = 1 max 2 \leq y \leq x max ψ (y; q, a) - \frac{y}{φ ( q )} ≪_{θ, A} \frac{x}{( lo g x ) ^{A}} (x \to \infty) .

NT-LFUNC-033

Artin Holomorphy Conjecture

LandmarkConjecture1923

Let

L / K

be a finite Galois extension of number fields with group

G

, and let

ρ : G \to GL (V)

be a nontrivial irreducible finite-dimensional complex representation. For every nonzero prime ideal

p

K

, choose a prime of

L

above it, let

I_{p}

be its inertia group, and let

Frob_{p}

denote an arithmetic Frobenius element acting on

V^{I_{p}}

. The Artin

L

-function

L (s, ρ, L / K) = p \prod det (1 - ρ (Frob_{p}) N p^{- s} ∣ V^{I_{p}})^{- 1}, ℜ s > 1,

has an analytic continuation to an entire function on

C

NT-ZERO-034

Montgomery’s Pair-Correlation Conjecture

LandmarkConjecture1973

Assume the Riemann hypothesis, write the nontrivial zeros of the Riemann zeta function as

\frac{1}{2} + iγ

, counted with multiplicity, and put

N (T) = # {γ : 0 < γ \leq T}

. For every fixed

0 < α < β

\frac{1}{N ( T )} # {(γ, γ^{'}) : 0 < γ, γ^{'} \leq T, α \leq \frac{( γ - γ ^{'} ) lo g T}{2 π} \leq β} ⟶ \int_{α}^{β} (1 - (\frac{sin π u}{π u})^{2}) d u

T \to \infty

NT-CORR-035

Chowla’s Correlation Conjecture

LandmarkConjecture1965

Let

λ (n) = (- 1)^{Ω (n)}

be the Liouville function, where

Ω (n)

is the number of prime factors of

n

, counted with multiplicity. For every integer

k \geq 2

and every choice of distinct nonnegative integers

h_{1}, \dots, h_{k}

n \leq x \sum j = 1 \prod k λ (n + h_{j}) = o (x) (x \to \infty) .

NT-DIO-036

$abc$ Conjecture

LandmarkConjecture1985

For every

ε > 0

, there is a constant

K_{ε} > 0

such that, whenever coprime positive integers

a, b, c

satisfy

a + b = c

c < K_{ε} rad (ab c)^{1 + ε}, rad (n) = p ∣ n \prod p .

NT-AN-004

Lindelöf Hypothesis

MajorConjecture1908

Let

ζ (s) = \sum_{n \geq 1} n^{- s}

for

ℜ s > 1

, and let

ζ

also denote its meromorphic continuation to

C

. For every

ε > 0

ζ (\frac{1}{2} + i t) = O_{ε} ((1 + ∣ t ∣)^{ε}) as ∣ t ∣ \to \infty.

NT-PRIME-007

Landau’s $n^2+1$ Prime Conjecture

MajorConjecture1912

There are infinitely many positive integers

n

for which

n^{2} + 1

is prime.

NT-PRIME-008

Legendre’s Conjecture

MajorConjecture1798

For every integer

n \geq 1

, there is a prime

p

such that

n^{2} < p < (n + 1)^{2}

NT-PRIME-010

Artin’s Primitive Root Conjecture

MajorConjecture1927

Let

g \in Z ∖ {- 1, 0, 1}

be not a perfect square. For

n \geq 1

, let

ζ_{n} = e^{2 πi / n}

, let

μ (n)

be the Möbius function (zero if a prime square divides

n

, and

(- 1)^{r}

n

is a product of

r

distinct primes), and put

δ (g) = n = 1 \sum \infty \frac{μ ( n )}{[ Q ( ζ _{n} , g ^{1/ n} ) : Q ]} .

Then

δ (g) > 0

and, as

x \to \infty

# {p \leq x : p prime, p ∤ g, g generates (Z / p Z)^{\times}} \sim δ (g) Li (x),

where

Li (x) = \int_{2}^{x} d t / lo g t

; the field in the denominator is independent of the choice of the

n

-th root of

g

NT-DIO-011

Fermat–Catalan Conjecture

MajorConjecture1995

There are only finitely many sextuples

(a, b, c; m, n, k) \in Z_{> 0}^{3} \times Z_{\geq 2}^{3}

such that

a^{m} + b^{n} = c^{k}, g cd (a, b, c) = 1, \frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1.

NT-DIO-012

Beal Conjecture

MajorConjecture1993

If positive integers

A, B, C, x, y, z

satisfy

A^{x} + B^{y} = C^{z}, x, y, z > 2,

then

g cd (A, B, C) > 1

NT-DIO-013

Erdős–Straus Conjecture

MajorConjecture1948

For every integer

n \geq 2

, there exist positive integers

x, y, z

such that

\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} .

NT-MULT-015

Odd Perfect Number Conjecture

MajorConjectureUnknown

There is no odd positive integer

n

satisfying

σ (n) = 2 n

, where

σ (n) = \sum_{d ∣ n, d > 0} d

NT-PRIME-016

Infinitude of Mersenne Primes

MajorConjectureUnknown

There are infinitely many primes

p

such that

2^{p} - 1

is prime.

NT-MULT-017

Lehmer’s Totient Conjecture

MajorConjecture1932

For

n \geq 1

, let

φ (n) = # (Z / n Z)^{\times}

be Euler’s totient. If

n > 1

and

φ (n)

divides

n - 1

, then

n

is prime.

NT-TRANS-018

Lehmer’s Mahler-Measure Conjecture

MajorExact constant problem1933

There exists a real constant

c > 1

such that every algebraic integer

α \neq = 0

that is not a root of unity satisfies

M (α) \geq c,

where, if

a_{0} \prod_{j = 1}^{d} (x - α_{j}) \in Z [x]

is the minimal polynomial of

α

, then

M (α) = ∣ a_{0} ∣ \prod_{j = 1}^{d} max (1, ∣ α_{j} ∣)

NT-MOD-019

Lehmer’s Nonvanishing Conjecture for Ramanujan’s $\tau$ -function

MajorConjecture1947

Define integers

τ (n)

n = 1 \sum \infty τ (n) q^{n} = q m = 1 \prod \infty (1 - q^{m})^{24} (∣ q ∣ < 1) .

Then

τ (n) \neq = 0

for every integer

n \geq 1

NT-DIOAPP-020

Littlewood Conjecture

MajorConjecturec. 1930

For all real numbers

α, β

n \to \infty lim inf n ∥ n α ∥ ∥ n β ∥ = 0,

where

∥ x ∥ = min_{m \in Z} ∣ x - m ∣

NT-TRANS-022

Four Exponentials Conjecture

MajorConjecturec. 1966

x_{1}, x_{2} \in C

are linearly independent over

Q

and

y_{1}, y_{2} \in C

are linearly independent over

Q

, then at least one of the four numbers

e^{x_{i} y_{j}} (1 \leq i, j \leq 2)

is transcendental.

NT-IWAS-023

Leopoldt’s Conjecture

MajorConjecture1962

Let

K

be a number field and

p

a rational prime. For each place

v ∣ p

, let

K_{v}

be the completion and let

lo g_{v} : O_{K_{v}}^{\times} \to K_{v}

be the local

p

-adic logarithm: if

m_{v}

is the maximal ideal of

O_{K_{v}}

, its restriction to

1 + m_{v}

lo g_{v} (1 + x) = m = 1 \sum \infty \frac{( - 1 ) ^{m + 1} x ^{m}}{m} (x \in m_{v}),

and it vanishes on torsion units. Then the map

O_{K}^{\times} \otimes_{Z} Q_{p} ⟶ v ∣ p \prod K_{v}, u ⟼ (lo g_{v} u)_{v ∣ p},

is injective.

NT-CYC-024

Vandiver’s Conjecture

MajorConjecturec. 1929

For every prime

p

, let

ζ_{p} = e^{2 πi / p}

and let

h_{p}^{+}

be the ideal-class number of the maximal real subfield

Q (ζ_{p} + ζ_{p}^{- 1})

of the

p

-th cyclotomic field. Then

p ∤ h_{p}^{+}

NT-CONST-026

Irrationality of the Euler–Mascheroni Constant

MajorOpen problemUnknown

The number

γ = n \to \infty lim (k = 1 \sum n \frac{1}{k} - lo g n)

is irrational.

NT-LATTICE-027

Gauss Circle-Problem Exponent Conjecture

MajorExact constant problemc. 1915

For

R \geq 1

, let

P (R) = # {(m, n) \in Z^{2} : m^{2} + n^{2} \leq R^{2}} - π R^{2} .

Then for every

ε > 0

P (R) = O_{ε} (R^{1/2 + ε}) (R \to \infty) .

NT-AN-028

Dirichlet Divisor-Problem Exponent Conjecture

MajorExact constant problemc. 1916

Let

d (n) = # {a \in Z_{> 0} : a ∣ n}

and

γ = lim_{N \to \infty} (\sum_{k = 1}^{N} 1/ k - lo g N)

, and put

Δ (x) = n \leq x \sum d (n) - x lo g x - (2 γ - 1) x .

For every

ε > 0

Δ (x) = O_{ε} (x^{1/4 + ε}) (x \to \infty) .

NT-SPECTRAL-029

Selberg Eigenvalue Conjecture

MajorConjecture1965

Let

Γ \leq SL_{2} (Z)

be a congruence subgroup, meaning that

Γ

contains

{g \in SL_{2} (Z) : g \equiv I (mod N)}

for some

N \geq 1

. If

λ > 0

is a discrete

L^{2}

-eigenvalue of the positive hyperbolic Laplacian

- y^{2} (\partial_{x}^{2} + \partial_{y}^{2})

Γ\ H

, where

H = {x + i y \in C : y > 0}

, then

λ \geq \frac{1}{4}

ALG-GALOIS-002

Inverse Galois Conjecture over $\mathbb Q$

LandmarkOpen problemc. 1892

For every finite group

G

, there exists a finite Galois extension

L / Q

such that

Gal (L / Q) ≅ G

ALG-MODREP-019

Brauer’s $k(B)$ -Conjecture

MajorConjecture1946

Let

G

be a finite group, let

p

be a prime, and fix a splitting

p

-modular system for

G

. Let

B

be a

p

-block of

G

, meaning a two-sided summand of the modular group algebra cut out by a primitive central idempotent; let

Irr (B)

be the set of irreducible ordinary complex characters belonging to

B

, and let

D \leq G

be a defect group of

B

. Then

k (B) := # Irr (B) \leq ∣ D ∣.

AG-CYCLE-003

Tate Conjecture

LandmarkConjecture1963

Let

k

be a finitely generated field,

X / k

a smooth projective variety,

r \geq 0

, and

ℓ \neq = char k

a prime. The image of

cl_{ℓ} : CH^{r} (X) \otimes_{Z} Q_{ℓ} ⟶ H_{\overset{e}{ˊ} t}^{2 r} (X_{\overset{ˉ}{k}}, Q_{ℓ} (r))

equals the invariant subspace

H_{\overset{e}{ˊ} t}^{2 r} (X_{\overset{ˉ}{k}}, Q_{ℓ} (r))^{Gal (\overset{ˉ}{k} / k)},

where

CH^{r} (X)

is the group of codimension-

r

algebraic cycles modulo rational equivalence.

AG-ANAB-005

Grothendieck Section Conjecture

LandmarkConjecture1983

Let

K

be a field finitely generated over

Q

, put

G_{K} = Gal (\overset{ˉ}{K} / K)

, and let

X / K

be a smooth, projective, geometrically connected curve of genus

g \geq 2

. After choosing a geometric base point, its étale fundamental groups fit into the exact sequence

1 \to π_{1}^{\overset{e}{ˊ} t} (X_{\overset{ˉ}{K}}) \to π_{1}^{\overset{e}{ˊ} t} (X) \to G_{K} \to 1.

The map sending

x \in X (K)

to the conjugacy class (under

π_{1}^{\overset{e}{ˊ} t} (X_{\overset{ˉ}{K}})

) of the section

G_{K} \to π_{1}^{\overset{e}{ˊ} t} (X)

determined by

x

is a bijection.

AG-DIO-006

Bombieri–Lang Conjecture

LandmarkConjecturec. 1971

K

is a number field and

X / K

is a smooth projective variety of general type, then

X (K)

is not Zariski dense in

X

AG-DIO-007

Vojta’s Main Conjecture

LandmarkConjecture1987

Let

X

be a smooth projective variety over a number field

K

, let

D

be a simple-normal-crossings divisor,

A

a big divisor,

S

a finite set of places of

K

r \geq 1

, and

ε > 0

. There is a proper Zariski-closed

Z \subset X

containing the support of

D

such that for every

P \in X (\overset{ˉ}{K}) ∖ Z

with

[K (P) : K] \leq r

m_{D, S} (P) + h_{K_{X}} (P) \leq d_{K} (P) + ε h_{A} (P) + O (1) .

Here

m_{D, S}

is the proximity function, heights use fixed Weil-height choices, the

O (1)

constant may depend on the fixed data

(X, D, A, K, S, r, ε)

but is independent of

P

, and

d_{K} (P) = \frac{lo g N _{K / Q} D _{K (P) / K}}{[ K ( P ) : K ]}

is the normalized logarithmic relative discriminant, with

D_{K (P) / K}

the relative discriminant ideal.

AG-DIFF-021

Grothendieck–Katz $p$ -Curvature Conjecture

LandmarkConjecture1969–1972

Let

K

be a number field, let

X / K

be a smooth connected variety, and let

(E, \nabla)

be an algebraic vector bundle with integrable connection on

X

. After extending these data over

O_{K} [S^{- 1}]

for some finite set

S

of finite places, reduce at a place

p \in / S

of residue characteristic

p

. For a local derivation

D

on the reduction

X_{p}

, define the

p

-curvature by

ψ_{p} (D) = \nabla_{D}^{p} - \nabla_{D^{[p]}},

where

D^{[p]}

is the

p

-fold iterate of

D

as a derivation. If

ψ_{p} (D) = 0

for every local derivation

D

and all but finitely many

p

, then, after any embedding

K ↪ C

, the analytic monodromy of

(E, \nabla)

X (C)

has finite image; equivalently, the connection becomes trivial after a finite étale cover of

X_{C}

AG-UNLIKELY-008

Zilber–Pink Conjecture for Abelian Varieties

MajorConjecture1999–2005

Let

A

be a complex abelian variety. For

s \geq 0

, let

A^{[s]}

be the union of all torsion cosets

τ + B

, where

τ \in A

is a torsion point and

B \subseteq A

is an abelian subvariety with

codim_{A} B \geq s

. If

X \subseteq A

is an irreducible closed subvariety not contained in any proper torsion coset, then

X \cap A^{[d i m X + 1]}

is not Zariski dense in

X

DYN-009

Furstenberg $\times2,\times3$ measure conjecture

LandmarkConjecture1967

Let

μ

be a Borel probability measure on

T = R / Z

that is invariant under

T_{2} (x) = 2 x (mod 1)

and

T_{3} (x) = 3 x (mod 1)

, and ergodic for their joint action: every Borel set invariant modulo

μ

under both maps has measure

0

1

. Then either

μ

is Lebesgue measure or there is a finite set

F \subset T

, invariant under both

T_{2}

and

T_{3}

, such that

μ (F) = 1

DYN-014

Sarnak Möbius disjointness conjecture

LandmarkConjecture2010

Define the Möbius function by

μ (1) = 1

μ (n) = 0

if a prime square divides

n

, and

μ (n) = (- 1)^{k}

n

is a product of

k

distinct primes. For every compact metric space

X

, every continuous map

T : X \to X

with zero topological entropy, every

f \in C (X; C)

, and every

x \in X

\frac{1}{N} n = 1 \sum N μ (n) f (T^{n} x) ⟶ 0 as N \to \infty.

COMB-004

Lonely Runner Conjecture

MajorConjecture1967

For every integer

n \geq 2

and every set of distinct positive integers

v_{1}, \dots, v_{n - 1}

, there exists

t \in R

such that

∥ t v_{i} ∥_{R / Z} \geq \frac{1}{n} (1 \leq i \leq n - 1),

where

∥ x ∥_{R / Z} = min_{z \in Z} ∣ x - z ∣

DG-005

Erdős–Ulam Rational-Distance Problem

MajorOpen problem1946

Does there exist a countable dense set

S \subset R^{2}

such that

∥ x - y ∥_{2} \in Q for every x, y \in S ?

TCS-012

Deterministic Polynomial-Time Integer Factoring

LandmarkOpen problemc. 1975

Is there a deterministic algorithm that, on binary input

N \geq 2

, outputs the prime factorization of

N

in time polynomial in

lo g N

LOG-DEC-005

Decidability of the Real Exponential Field

LandmarkOpen problemc. 1950

There exists an algorithm which, given any first-order sentence

φ

in the language

{0, 1, +, \cdot, <, exp}

, halts and correctly decides whether

(R; 0, 1, +, \cdot, <, x \mapsto e^{x}) ⊨ φ .