Maths WorkshopMaths problems with full solutions

Number theory problems

287 problems · Divisibility, primes, congruences and irrationality proofs.

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Ex.11Product of irrational and nonzero rational

#irrationality #product #rational

Ex.12Sum of irrational and rational

#irrationality #sum #rational

Ex.23Irrationality of √2 + √3

#irrational #proof by contradiction

Ex.36Irrationality of ∛2

#cube root #irrationality #unique factorization

Ex.37Between any two rationals lies an irrational

#density #irrationals #betweenness

Ex.38Irrationality of √p for prime p

#sqrt p #prime #irrationality

Ex.39Irrationality of √2

#sqrt2 #irrationality #proof by contradiction

Ex.40Irrationality of √3

#sqrt3 #irrationality #mod 3

Ex.41Linear independence of logs

#logarithm #independence #irrationality

Ex.57Irrationality of ⁿ√2

#nth root #irrationality #prime factorization

Ex.58Irrationality of log₁₀ 2

#logarithm #irrationality #base 10

Ex.59Irrationality of log₂ 3

#logarithm #irrationality #base change

Ex.60A square root in disguise

#irrationality #quadratic #conjugates

Ex.61Ratios involving irrationals

#ratio #irrationality #algebraic

Ex.71Decimal expansions and rationality

#decimal expansion #rational #periodic

Ex.72Irrationality of a cubic root expression

#cubic root #irrationality #algebraic

Ex.73Irrationality of √2 + √3

#algebraic #sum of roots #irrationality

Ex.74A sum of radicals

#radicals #irrationality #algebraicity

Ex.84Rational root theorem and irrationality

#rational root theorem #irrationality #cubic

Ex.89Sum of irrationals in disguise

#irrationality #sum #structure

Ex.90Irrationality of π

#pi #irrationality #Lambert

Ex.91Continued fractions and irrationality

#continued fractions #irrationality #sqrt2

Ex.92Irrationality of √n for non-square n

#sqrt n #non-square #irrationality

Ex.95x³ + 2y³ + 4z³ = 0

#infinite descent #parity #irrationality

Ex.98Liouville numbers

#Liouville #transcendental #irrationality

Ex.99A general irrationality criterion

#criterion #irrationality #polynomial

Ex.115Gelfond–Schneider theorem (statement)

#Gelfond-Schneider #transcendence #Hilbert 7th

Ex.122Fractional parts of multiples of √2

#fractional parts #density #irrational

Ex.204Basic divisibility and linear combinations

#divisibility #linear combinations #divisors

Divisibility and gcd

Ex.205Consecutive integers are coprime

#coprime #consecutive integers #Euclidean algorithm

Divisibility and gcd

Ex.206Finding divisors systematically

#divisors #enumeration #bounds

Divisibility and gcd

Ex.223Arithmetic modulo $n$

#modular arithmetic #operations #compatibility

Ex.224Congruence as an equivalence relation

#equivalence relation #congruence #residue classes

Ex.232Divisibility rules

#divisibility rules #modular arithmetic #base 10

Divisibility and gcd

Ex.233GCD of a GCD

#gcd #associativity #three numbers

Divisibility and gcd

Ex.234The HCF of consecutive integers

#coprime #consecutive integers #gcd

Divisibility and gcd

Ex.247Sum of irrationals can be rational

#irrationality #sum #counterexample

Ex.262GCD of successive integers

#coprime #consecutive integers #gcd

Ex.263Distinct primes are coprime

#coprime #primes #distinct

Ex.264Definition and first examples of primes

#primes #definition #examples

Ex.265GCD with a prime

#gcd #prime #divisor

Ex.317A large power modulo 7

#congruences #periodicity

Ex.325The Euclidean algorithm and Bézout

Divisibility and gcd

Ex.326There are infinitely many primes

#primes #proof by contradiction

Ex.3346 divides n³ − n

#divisibility #factorisation

Divisibility and gcd

Ex.338Sums of two squares modulo 4

#congruences #squares

Ex.342The last digit of 7^2026

#congruences #periodicity

Ex.366Common divisors and common multiples

#divisibility #lcm #structure

Divisibility and gcd

Ex.367Computing LCM from divisibility

#lcm #divisibility #product

Divisibility and gcd

Ex.368$n^2-1$ is divisible by 8 for odd $n$

#divisibility #parity #consecutive integers

Divisibility and gcd

Ex.369The relation between divisibility and lcm

#divisibility #lcm #product relation

Divisibility and gcd

Ex.370Simplifying a fraction with the divisibility

#divisibility #irreducible fraction #simplification

Divisibility and gcd

Ex.371divisibility of three numbers

#divisibility #three numbers #prime factorization

Divisibility and gcd

Ex.434A linear congruence

#linear congruence #inverse #solution

Ex.453Coprime integers

#coprime #gcd #Bézout

Divisibility and gcd

Ex.454GCD of three numbers

#gcd #three numbers #Euclidean

Divisibility and gcd

Ex.455Congruence classes modulo $n$

#congruences #residue classes #partition

Divisibility and gcd

Ex.456GCD and divisibility

#gcd #divisibility #equivalence

Divisibility and gcd

Ex.457LCM of three numbers

#lcm #three numbers #prime factorization

Divisibility and gcd

Ex.458The least common multiple

#lcm #gcd #product relation

Divisibility and gcd

Ex.459Every integer has a prime divisor

#primes #divisor #well-ordering

Divisibility and gcd

Ex.493Between any two reals lies a rational

#density #rationals #betweenness

Ex.494√2 is algebraic

#algebraic #sqrt2 #minimal polynomial

Ex.541Consecutive primes and coprimality

#consecutive primes #coprime #primorial

Ex.542Coprimality with primes

#coprime #prime #divisibility

Ex.543The fundamental theorem of arithmetic

#fundamental theorem #factorization #uniqueness

Ex.544GCD and LCM via prime factorization

#gcd #lcm #prime factorization

Ex.545Primes in GCD and LCM

#gcd #lcm #prime factorization

Ex.546Goldbach's conjecture (statement and verification)

#Goldbach conjecture #even numbers #primes

Ex.547Euclid's proof of infinitely many primes

#Euclid #contradiction #primes

Ex.548Iterated GCD

#gcd #iteration #associativity

Ex.549Prime factorization and divisors

#prime factorization #divisors #exponents

Ex.550Characterization of primes

#primes #characterization #divisors

Ex.551Using the sieve to test primality

#sieve #primality #application

Ex.552The sieve of Eratosthenes

#sieve #Eratosthenes #primes

Ex.553Smallest prime divisor

#smallest divisor #compositeness #sqrt bound

Ex.554The smallest prime factor

#smallest prime factor #compositeness #bounds

Ex.555Trial division primality test

#trial division #primality test #algorithm

Ex.556Twin primes

#twin primes #examples #conjecture

Ex.618n + S(n) + S(S(n)) = 60

#digit sum #congruence modulo 9 #bounding

Ex.6198k + 7 is not a sum of three squares

#squares modulo 8 #representation by squares

Ex.620Counting divisors and the hundred lockers

#divisor function #perfect squares #pairing

Divisibility and gcd

Ex.621x² + y² = 3z² has no non-trivial solution

#infinite descent #squares modulo 3 #Diophantine equations

Diophantine equations

Ex.622x² − y² = 2024

#Diophantine equations #factoring #parity #divisors

Diophantine equations

Ex.623gcd(n! + 1, (n + 1)! + 1) = 1

#gcd #divisibility #factorials

Divisibility and gcd

Ex.62413 divides 2⁷⁰ + 3⁷⁰

#Fermat's little theorem #modular exponentiation

Divisibility and gcd

Ex.625When can 2ⁿ + 1 or 2ⁿ − 1 be prime?

#factorisation #primes #Fermat primes #Mersenne primes

Ex.643When is 1! + 2! + … + n! a perfect square?

#squares modulo n #factorials #Diophantine

Diophantine equations

Ex.644Writing 1 as a sum of unit fractions

#Egyptian fractions #bounding #Diophantine

Diophantine equations

Ex.645The sum of φ(d) over the divisors of n

#Euler totient #divisors #counting #fractions

Divisibility and gcd

Ex.70630 divides n⁵ − n

#divisibility #factorisation

Divisibility and gcd

Ex.740A divisibility identity for a sum

#divisibility #linearity #sum

Divisibility and gcd

Ex.741Bézout identity for 1001 and 315

#Euclidean algorithm #Bézout identity

Divisibility and gcd

Ex.742When are $2n$ and $3n+1$ coprime?

#divisibility #linear combination #coprimality

Divisibility and gcd

Ex.743Coprime powers remain coprime

#coprime #powers #Bézout

Divisibility and gcd

Ex.744If $a,b$ are coprime, so are $a+b$ and $ab$

#coprime #sum #product

Divisibility and gcd

Ex.745Divisibility of $a^n-b^n$

#divisibility #factorization #powers

Divisibility and gcd

Ex.746Divisors of a product

#divisibility #coprime #product divisibility

Divisibility and gcd

Ex.747Why the Euclidean algorithm works

#Euclidean algorithm #proof #division

Divisibility and gcd

Ex.748The Euclidean algorithm and Bézout identity

#Euclidean algorithm #Bézout identity #divisibility computation

Divisibility and gcd

Ex.749When is $(6n+12)/(2n+1)$ an integer?

#divisibility #fraction #case analysis

Divisibility and gcd

Ex.750An incompatible congruence system

#Chinese remainder #compatibility condition

Ex.751When does $n-4$ divide $3n-17$?

#divisibility #linear combination #case analysis

Divisibility and gcd

Ex.752When is the number of divisors odd?

#divisors #perfect squares #counting

Divisibility and gcd

Ex.753A divisibility recurrence

#divisibility #recurrence #Euclidean algorithm

Divisibility and gcd

Ex.754The smallest prime divisor

#primes #smallest divisor #compositeness

Divisibility and gcd

Ex.841Coprimality via congruences

#coprime #congruence #linear combination

Ex.842A CRT application

#CRT #application #calendar

Ex.843The Chinese remainder theorem (two moduli)

#Chinese remainder theorem #CRT #coprime moduli

Ex.844Euler's theorem

#Euler theorem #totient #modular exponentiation

Ex.845Computing $\varphi(n)$

#Euler totient #computation #prime factorization

Ex.846Generalization of Fermat

#Fermat #generalization #coprime

Ex.847Fermat's little theorem

#Fermat #prime #modular exponentiation

Ex.848Halving in modular arithmetic

#halving #modular arithmetic #odd modulus

Ex.849Finding modular inverses

#modular inverse #extended Euclidean #algorithm

Ex.850Inverse via Fermat

#Fermat #inverse #modular arithmetic

Ex.851Solvability of $ax\equiv b\pmod n$

#linear congruence #gcd criterion #number of solutions

Ex.852Applications of the modular inverse

#modular inverse #application #linear congruence

Ex.853Power cycles modulo $n$

#power cycle #periodicity #modulus

Ex.854A system of congruences

#system #CRT #linear congruences

Ex.855The order of an element mod $n$

#order #multiplicative group #divisor

Ex.856Computing large powers mod $n$

#modular exponentiation #square-and-multiply #fast power

Ex.908Bézout's identity

#Bézout #Euclidean algorithm #linear combination

Divisibility and gcd

Ex.909Coprime product

#coprime #product #divisibility

Divisibility and gcd

Ex.910Infinitely many primes

#Euclid #primes #proof by contradiction

Divisibility and gcd

Ex.911The fundamental theorem of arithmetic

#fundamental theorem #prime factorization #uniqueness

Divisibility and gcd

Ex.912Gauss's lemma

#Gauss lemma #divisibility #coprime

Divisibility and gcd

Ex.913GCD and product of coprime factors

#gcd #coprime #factorization

Divisibility and gcd

Ex.914Divisors of coprime numbers

#coprime #divisors #factorization

Divisibility and gcd

Ex.915Linear relations and coprimality

#coprime #linear relation #proof

Divisibility and gcd

Ex.916Proof of the Euclidean algorithm

#Euclidean algorithm #gcd invariance #proof

Divisibility and gcd

Ex.917GCD as a linear combination

#gcd #linear combination #Bézout

Divisibility and gcd

Ex.918GCD and linear combinations

#gcd #linear combination #divisors

Divisibility and gcd

Ex.919GCD of powers

#gcd #powers #prime factorization

Divisibility and gcd

Ex.920A coprimality proof

#coprime #proof #linear combination

Divisibility and gcd

Ex.921A recursive GCD

#gcd #Fibonacci #recursion

Divisibility and gcd

Ex.922Semiprimes and divisors

#semiprime #divisors #factorization

Divisibility and gcd

Ex.923Additive property of the p-adic valuation

#p-adic valuation #additivity #non-Archimedean

Divisibility and gcd

Ex.924Finding modular inverses

#modular inverse #Bézout #congruences

Divisibility and gcd

Ex.925Number of divisors

#divisors #prime factorization #counting

Divisibility and gcd

Ex.926Perfect numbers

#perfect numbers #sum of divisors #Euclid

Divisibility and gcd

Ex.927Perfect squares and divisors

#perfect squares #divisors #parity

Divisibility and gcd

Ex.928Prime power divisibility

#prime power #valuation #p-adic

Divisibility and gcd

Ex.929The radical of an integer

#radical #squarefree #prime factorization

Divisibility and gcd

Ex.930Sum of divisors

#sum of divisors #multiplicative #formula

Divisibility and gcd

Ex.931Units modulo $n$

#units #invertible #Euler totient

Divisibility and gcd

Ex.932p-adic valuation

#p-adic valuation #divisibility #properties

Divisibility and gcd

Ex.1026Rational root theorem and irrationality

#rational root theorem #algebraic #irrationality

Ex.1027Uncountability of irrationals

#uncountability #Cantor #irrationals

Ex.1028Rational coefficients and irrational roots

#rational coefficients #irrational roots #conjugates

Ex.1029Degree of algebraic numbers

#algebraic degree #minimal polynomial #sqrt2

Ex.1030Rational, algebraic, transcendental

#hierarchy #algebraic #transcendental

Ex.1118Coprimality and Euler's theorem

#coprime #Euler theorem #application

Ex.1119Coprimality with factorials

#coprime #factorial #prime

Ex.1120Factorial and coprime integers

#factorial #coprime #bounds

Ex.1121Consecutive primes and products

#consecutive primes #product #gap

Ex.1122Coprime integers and prime divisors

#coprime #prime divisors #intersection

Ex.1123Coprime products and primes

#coprime #product #prime

Ex.1124Divisibility by coprime integers

#coprime #divisibility #product

Ex.1125Coprime remainders

#coprime #remainders #CRT

Ex.1126Multiplicativity of $\varphi$

#Euler totient #multiplicative #CRT

Ex.1127Euler's totient of a prime power

#Euler totient #prime power #formula

Ex.1128Factoring a large integer

#factorization #primes #algorithm

Ex.1129Fermat's little theorem

#Fermat #prime #modular exponentiation

Ex.1130Fermat primality test

#Fermat test #primality #witness

Ex.1131Fermat's little theorem and compositeness

#Fermat #compositeness test #modular exponentiation

Ex.1132Totient and coprime residue classes

#totient #coprime #residue classes

Ex.1133Euclid's lemma

#Euclid lemma #prime #divisibility

Ex.1134Primes and factorials

#factorial #primes #Wilson

Ex.1135GCD of a product

#gcd #product #prime factorization

Ex.1136Mersenne primes

#Mersenne #primes #perfect numbers

Ex.1137Mersenne primes and exponents

#Mersenne #primes #exponents

Ex.1138Order modulo a prime

#order #prime #divisor

Ex.1139Even perfect numbers and Mersenne primes

#perfect numbers #Mersenne #Euclid-Euler

Ex.1140The prime counting function

#prime counting #pi function #estimates

Ex.1141Prime gaps

#prime gaps #arbitrary gaps #construction

Ex.1142Prime powers and divisibility

#prime powers #divisibility #exponents

Ex.1143Primality test up to $\sqrt n$

#primality test #sqrt #trial division

Ex.1144Sophie Germain primes

#Sophie Germain #primes #examples

Ex.1277Long runs of non-squarefree numbers

#Chinese remainder theorem #squarefree numbers

Divisibility and gcd

Ex.1278Infinitely many primes of the form 4k + 3

#primes #Euclid's argument #residues

Ex.1279n divides 2ⁿ − 1 only for n = 1

#order #Fermat's little theorem #smallest prime divisor

Ex.1280The gcd of 2ᵃ − 1 and 2ᵇ − 1

#gcd #Euclidean algorithm #divisibility

Divisibility and gcd

Ex.128160 divides xyz for every Pythagorean triple

#Pythagorean triples #residues #squares modulo n

Divisibility and gcd

Ex.1282Wilson's theorem and its converse

#Wilson's theorem #modular inverses #primality

Ex.1328A multiple of n written with only 0s and 1s

#pigeonhole #repunits #decimal digits

Ex.1329(n²)! is divisible by (n!)^(n+1)

#integrality #factorials #double counting #multinomial

Divisibility and gcd

Ex.1330If aᵖ ≡ bᵖ (mod p) then aᵖ ≡ bᵖ (mod p²)

#Fermat's little theorem #binomial theorem #lifting

Ex.1331Infinitely many n dividing 2ⁿ + 1

#induction #lifting the exponent #infinitely many #factorisation

Divisibility and gcd

Ex.1332Infinitely many primes ≡ 1 (mod 4)

#infinitely many primes #order #Euclid's argument

Ex.1333v_p(n!) from the digits of n in base p

#Legendre's formula #digit sum #p-adic valuation #base p

Divisibility and gcd

Ex.1334Counting odd entries in a row of Pascal's triangle

#binomial coefficients #parity #binary expansion #generating function

Ex.1335Prime divisors of 2ᵖ − 1

#Mersenne numbers #order #primality testing

Ex.1336Prime divisors of n² + n + 1

#order #Fermat's little theorem #primes in progressions

Ex.1337((p − 1)/2)! modulo a prime p ≡ 3 (mod 4)

#Wilson's theorem #factorials modulo p #quadratic residues

Ex.1338The Frobenius coin problem for two coins

#Frobenius number #linear Diophantine #Chicken McNugget

Diophantine equations

Ex.1339The sum of the quadratic residues modulo p

#quadratic residues #sum of squares #divisibility

Ex.1340Infinitely many triangular numbers that are squares

#Pell equation #recurrence #infinitely many

Diophantine equations

Ex.1482Existence of a Bézout identity

#Bézout #existence #well-ordering

Divisibility and gcd

Ex.1483A Chinese remainder system

#Chinese remainder theorem #congruences #Bézout

Ex.1484A common divisor divides 43

#divisibility #linear combination #prime number

Divisibility and gcd

Ex.1485Solving a linear Diophantine equation

#Diophantine equation #Bézout #particular solution

Diophantine equations

Ex.1486Solving $ax+by=c$

#Diophantine equation #Bézout #solvability criterion

Diophantine equations

Ex.1487Solving $\divisibility(a,b)+\operatorname{lcm}(a,b)=9+b$

#divisibility #lcm #Diophantine equation

Divisibility and gcd

Ex.1488A prime divisor property

#primes #divisibility #Euclid's lemma

Divisibility and gcd

Ex.1578CRT with non-coprime moduli

#CRT #non-coprime #compatibility

Ex.1579The CRT as a ring isomorphism

#CRT #ring isomorphism #product

Ex.1580The CRT with three moduli

#CRT #three moduli #pairwise coprime

Ex.1581The Diffie–Hellman protocol

#Diffie-Hellman #cryptography #modular

Ex.1582Euler's theorem applied

#Euler theorem #totient #computation

Ex.1583Fermat's little theorem and pseudoprimes

#Fermat #pseudoprime #compositeness

Ex.1584Proving Fermat's little theorem

#Fermat #proof #permutation

Ex.1585Legendre symbol and Euler's criterion

#Legendre symbol #Euler criterion #quadratic reciprocity

Ex.1586RSA and modular arithmetic

#RSA #cryptography #modular exponentiation

Ex.1587Primitive roots

#primitive root #generator #cyclic group

Ex.1588Quadratic residues mod $p$

#quadratic residues #Legendre symbol #Euler criterion

Ex.1589A system of linear congruences

#system #linear congruences #CRT

Ex.1590Wilson's theorem

#Wilson theorem #prime #factorial

Ex.1644Egyptian fractions

#Egyptian fractions #unit fractions #greedy algorithm

Divisibility and gcd

Ex.1645Chinese remainder theorem

#Chinese remainder #congruences #CRT

Divisibility and gcd

Ex.1646Coprime numbers and exponents

#coprime #exponents #gcd

Divisibility and gcd

Ex.1647Coprime sums and products

#coprime #sum #product

Divisibility and gcd

Ex.1648An application of the Euclidean algorithm

#Euclidean algorithm #expansion #continued fraction

Divisibility and gcd

Ex.1649A Diophantine equation with coefficients

#Diophantine #Bézout #general solution

Divisibility and gcd

Ex.1650Euler's totient function

#Euler totient #coprime #formula

Divisibility and gcd

Ex.1651GCD of sums

#gcd #sums #congruence

Divisibility and gcd

Ex.1652GCD via the Euclidean matrix

#gcd #Euclidean algorithm #matrix

Divisibility and gcd

Ex.1653Divisors of Mersenne numbers

#Mersenne #divisors #congruences

Divisibility and gcd

Ex.1654GCD in $\mathbb{Z}$ vs $\mathbb{R}[X]$

#analogy #gcd #polynomials #integers

Divisibility and gcd

Ex.1655GCD of polynomial coefficients

#gcd #coefficients #polynomials

Divisibility and gcd

Ex.1656Primitive Pythagorean triples

#Pythagorean triples #coprime #parity

Divisibility and gcd

Ex.1752$e$ is not a quadratic irrational

#e #quadratic irrational #contradiction

Ex.1753Liouville's theorem and examples

#Liouville #approximation #transcendental

Ex.1754Independence of square roots

#independence #square roots #linear independence

Ex.1831Bertrand's postulate (statement and examples)

#Bertrand #primes #interval

Ex.1832Quadratic reciprocity applied

#quadratic reciprocity #Legendre #application

Ex.1833Primes of the form $4n+1$

#Dirichlet #arithmetic progression #primes

Ex.1834Fermat test detects compositeness

#Fermat test #witness #compositeness

Ex.1835Fermat factorization

#Fermat factorization #factoring #difference of squares

Ex.1836GCD via Euler's totient

#gcd #Euler totient #formula

Ex.1837Primorial and GCD

#primorial #gcd #primes

Ex.1838Bounds on the $n$-th prime

#n-th prime #bounds #elementary

Ex.1839Prime numbers and prime ideals

#prime ideal #Z #prime numbers

Ex.1840Finding primitive roots

#primitive root #mod prime #computation

Ex.1841Pseudoprimes

#pseudoprime #Fermat #compositeness

Ex.1842Primes and polynomial irreducibility

#irreducibility #primes #polynomial

Ex.1843The sieve of Sundaram

#sieve #Sundaram #primes

Ex.1957⌊√n + √(n+1)⌋ = ⌊√(4n+2)⌋

#floor function #square roots #squares modulo 4

Divisibility and gcd

Ex.1958Primes 3 mod 4 and sums of two squares

#sum of two squares #quadratic residues #descent

Ex.1959When is n⁴ + n³ + n² + n + 1 a perfect square?

#bounding between squares #Diophantine equations

Diophantine equations

Ex.1960An integrality proof by Legendre's formula

#Legendre's formula #floor function #integrality

Ex.1961Prime factors of Fermat numbers

#order #Fermat numbers #Fermat's little theorem

Divisibility and gcd

Ex.1962The exact power of 3 dividing 2^(3ⁿ) + 1

#p-adic valuation #lifting the exponent #induction

Ex.1995Even perfect numbers (Euclid–Euler)

#perfect numbers #sigma function #multiplicativity #Mersenne primes

Divisibility and gcd

Ex.1996Every prime ≡ 1 (mod 4) is a sum of two squares

#sum of two squares #Thue's lemma #pigeonhole #Wilson

Ex.1997n divides φ(aⁿ − 1)

#order #Euler's theorem #totient

Ex.1998Which primes divide C(2n, n)?

#Legendre's formula #central binomial coefficient #Bertrand's postulate

Ex.1999gcd(F_m, F_n) = F_gcd(m,n)

#Fibonacci numbers #gcd #Euclidean algorithm #divisibility sequences

Divisibility and gcd

Ex.2108A Carmichael number check

#Carmichael #Fermat #pseudoprime

Ex.2109The Carmichael function $\lambda(n)$

#Carmichael function #lcm #exponent

Ex.2110The discrete logarithm

#discrete logarithm #cyclic group #Pohlig-Hellman

Ex.2111Power tower modulo $n$

#power tower #modular exponentiation #Carmichael

Ex.2112Square roots modulo $p$

#square roots #mod p #Tonelli-Shanks

Ex.2113Stirling numbers via modular arithmetic

#Stirling #modular arithmetic #combinatorics

Ex.2114NTT and modular arithmetic (statement)

#NTT #number theory transform #FFT #modular

Ex.2122Carmichael numbers (statement)

#Carmichael #primes #Fermat

Divisibility and gcd

Ex.2123The Miller–Rabin test

#primality testing #Miller-Rabin #modular arithmetic

Divisibility and gcd

Ex.2173Algebraic independence of $e$ and $\pi$

#algebraic independence #e #pi

Ex.2174Irrationality of e+ π and πe

#irrationality #e #pi #open problem

Ex.2175Euler's number and transcendence

#e #transcendence #Euler

Ex.2176Schanuel's conjecture (statement)

#Schanuel #conjecture #transcendence

Ex.2191The AKS primality test (statement)

#AKS #deterministic primality #polynomial time

Ex.2192Elliptic curve factorization (statement)

#elliptic curve #factorization #Lenstra

Ex.2193The Fermat quotient

#Fermat quotient #prime #congruence

Ex.2194The Lucas–Lehmer test

#Lucas-Lehmer #Mersenne #primality

Ex.2195Chebyshev's bounds

#Chebyshev #bounds #prime counting

Ex.2196Primes in arithmetic progressions

#Dirichlet theorem #arithmetic progression #primes

Ex.2197The prime number theorem (statement)

#prime number theorem #asymptotic #pi function

Ex.2198Existence of primitive roots

#primitive root #existence #cyclic group

Ex.2199Proth's theorem

#Proth #primes #primality

Ex.2200Quadratic reciprocity (statement)

#quadratic reciprocity #Legendre symbol #primes

Ex.2201Sophie Germain's theorem

#Sophie Germain #Fermat #primes

Ex.2202Stirling's formula and primes

#Stirling #factorial #prime counting

Ex.2203The Miller–Rabin test

#Miller-Rabin #primality #probabilistic

Ex.2204Wieferich primes (statement)

#Wieferich #Fermat quotient #primes

Ex.2233Vieta jumping: (a² + b²)/(ab + 1) is a square

#Vieta jumping #descent #Diophantine equations #extremal principle

Diophantine equations

Ex.2240Primitive roots exist modulo every prime

#primitive root #order #totient #polynomial roots