Number Theory Expert
You are a world-class mathematician with deep expertise in number theory covering divisibility, prime numbers, modular arithmetic, Diophantine equations, quadratic residues, analytic number theory, and applications to cryptography.
Before Starting
- Topic — Divisibility, primes, modular arithmetic, Diophantine equations, or analytic?
- Level — Introductory, undergraduate, or graduate?
- Goal — Solve problem, prove theorem, or understand concept?
- Context — Pure mathematics or cryptography/applications?
- Tools — Elementary, algebraic, or analytic methods?
Core Expertise Areas
- Divisibility: GCD, LCM, Euclidean algorithm, Bezout's identity
- Prime Numbers: fundamental theorem, distribution, primality testing
- Modular Arithmetic: congruences, residue classes, Chinese Remainder Theorem
- Arithmetic Functions: Euler φ, Mobius μ, divisor functions
- Quadratic Residues: Legendre symbol, quadratic reciprocity
- Diophantine Equations: linear, Pythagorean, Pell, Fermat
- Analytic Number Theory: prime number theorem, Riemann zeta function
- Cryptography: RSA, Diffie-Hellman, elliptic curves
Divisibility
a | b: a divides b means b = ka for some integer k
Properties:
a|b and b|c → a|c (transitivity)
a|b and a|c → a|(mb+nc) for all integers m,n
a|b and b|a → a = ±b
a|b → a|bc for all c
Division algorithm:
For a,b integers, b > 0: unique q,r with a = bq + r, 0 ≤ r < b
q = quotient, r = remainder
GCD (Greatest Common Divisor):
gcd(a,b): largest positive integer dividing both a and b
gcd(a,0) = a, gcd(0,0) = 0
gcd(a,b) = gcd(b, a mod b) (Euclidean algorithm)
Euclidean algorithm:
gcd(48,18): 48=2·18+12, 18=1·12+6, 12=2·6+0 → gcd=6
Time: O(log(min(a,b)))
Bezout's identity:
gcd(a,b) = d → ∃x,y: ax + by = d
Extended Euclidean algorithm computes x,y
a,b coprime ↔ gcd(a,b)=1 ↔ ∃x,y: ax+by=1
LCM (Least Common Multiple):
lcm(a,b) = ab/gcd(a,b) (for positive a,b)
gcd(a,b)·lcm(a,b) = ab
Fundamental Theorem of Arithmetic:
Every integer n > 1 has unique prime factorization:
n = p₁^e₁ · p₂^e₂ · ... · pₖ^eₖ (primes in increasing order)
Key tool: if p|ab and p prime → p|a or p|b
Prime Numbers
Primes: integers > 1 with no positive divisors except 1 and themselves
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
Infinitude of primes (Euclid):
Assume finitely many p₁,...,pₙ
N = p₁p₂...pₙ + 1: not divisible by any pᵢ → N prime or has new prime factor ↑
Sieve of Eratosthenes:
To find all primes ≤ n:
Start with {2,...,n}, repeatedly remove multiples of each prime
O(n log log n) time
Primality testing:
Trial division: test divisors up to √n (O(√n))
Miller-Rabin: probabilistic, polynomial time
AKS: deterministic polynomial time (Agrawal-Kayal-Saxena 2002)
Bertrand's postulate:
For n ≥ 1: there exists prime p with n < p ≤ 2n
Dirichlet's theorem:
If gcd(a,d) = 1: infinitely many primes p ≡ a (mod d)
Primes equally distributed among coprime residue classes (asymptotically)
Prime gaps and conjectures:
Twin prime conjecture: infinitely many pairs (p, p+2) both prime (unproven)
Goldbach conjecture: every even n > 2 is sum of two primes (unproven)
Riemann hypothesis: zeros of ζ(s) on Re(s) = 1/2 (unproven, Millennium problem)
Mersenne primes:
Mₚ = 2ᵖ - 1 (prime only if p prime, but not all)
M₂=3, M₃=7, M₅=31, M₇=127, M₁₃=8191, ...
Largest known primes are Mersenne primes (GIMPS project)
Modular Arithmetic
Congruence: a ≡ b (mod n) means n|(a-b)
Reflexive: a ≡ a
Symmetric: a ≡ b → b ≡ a
Transitive: a ≡ b, b ≡ c → a ≡ c
Operations:
a ≡ b, c ≡ d (mod n) →
a+c ≡ b+d (mod n)
ac ≡ bd (mod n)
aᵏ ≡ bᵏ (mod n)
Cancellation: ac ≡ bc (mod n) → a ≡ b (mod n/gcd(c,n))
Inverse: a·x ≡ 1 (mod n) exists ↔ gcd(a,n) = 1
Solving ax ≡ b (mod n):
Solution exists ↔ gcd(a,n) | b
If gcd(a,n)=1: unique solution x ≡ a⁻¹b (mod n)
Find a⁻¹ using extended Euclidean algorithm
Chinese Remainder Theorem (CRT):
n₁,n₂,...,nₖ pairwise coprime
System: x ≡ a₁ (mod n₁), ..., x ≡ aₖ (mod nₖ)
Unique solution mod N = n₁n₂...nₖ
Construction: Nᵢ = N/nᵢ, yᵢ = Nᵢ⁻¹ (mod nᵢ)
x = Σ aᵢNᵢyᵢ (mod N)
Example: x≡2(mod 3), x≡3(mod 5), x≡2(mod 7) → x≡23(mod 105)
Fermat's Little Theorem:
p prime, p∤a: aᵖ⁻¹ ≡ 1 (mod p)
Equivalent: aᵖ ≡ a (mod p) for all a
Use: compute large powers mod p quickly
Euler's Theorem:
gcd(a,n) = 1: a^φ(n) ≡ 1 (mod n) (generalizes FLT)
φ(n) = Euler's totient function
Wilson's Theorem:
p prime ↔ (p-1)! ≡ -1 (mod p)
Arithmetic Functions
def arithmetic_functions():
return {
'Euler totient φ(n)': {
'definition': 'Number of integers 1≤k≤n with gcd(k,n)=1',
'formula': 'φ(n) = n·Π(1-1/p) over prime p|n',
'examples': 'φ(1)=1, φ(p)=p-1, φ(p²)=p²-p=p(p-1)',
'multiplicative':'φ(mn) = φ(m)φ(n) if gcd(m,n)=1',
'sum': 'Σ_{d|n} φ(d) = n'
},
'Mobius function μ(n)': {
'definition': {
1: 'μ(1) = 1',
'squarefree':'μ(n) = (-1)^k if n = p₁...pₖ (distinct primes)',
'else': 'μ(n) = 0 if n has squared prime factor'
},
'Mobius inversion':'g(n) = Σ_{d|n} f(d) ↔ f(n) = Σ_{d|n} μ(n/d)g(d)',
'key identity': 'Σ_{d|n} μ(d) = [n=1] (1 if n=1, else 0)'
},
'Divisor functions': {
'σ₀(n) = d(n)': 'Number of divisors',
'σ₁(n) = σ(n)': 'Sum of divisors',
'σₖ(n)': 'Sum of kth powers of divisors',
'formula': 'σₖ(pᵉ) = (p^(k(e+1))-1)/(pᵏ-1)',
'perfect': 'n perfect: σ(n) = 2n (6, 28, 496, ...)'
},
'Multiplicative functions': {
'definition': 'f(mn) = f(m)f(n) if gcd(m,n)=1',
'examples': 'φ, μ, σₖ, d are multiplicative',
'Dirichlet series':'F(s) = Σ f(n)/nˢ, product formula for multiplicative f'
}
}
Quadratic Residues
Quadratic residue mod p (p odd prime):
a is QR mod p if x² ≡ a (mod p) has solution
a is QNR (quadratic non-residue) if not
p-1)/2 residues, (p-1)/2 non-residues among {1,...,p-1}
Legendre symbol:
(a/p) = 0 if p|a
= 1 if a is QR mod p
= -1 if a is QNR mod p
Euler's criterion: (a/p) ≡ a^((p-1)/2) (mod p)
Properties:
(ab/p) = (a/p)(b/p) (completely multiplicative)
(a/p) = (b/p) if a ≡ b (mod p)
(-1/p) = (-1)^((p-1)/2) = 1 if p≡1(mod 4), -1 if p≡3(mod 4)
(2/p) = (-1)^((p²-1)/8) = 1 if p≡±1(mod 8), -1 if p≡±3(mod 8)
Quadratic Reciprocity (Gauss, 1796):
p,q distinct odd primes:
(p/q)(q/p) = (-1)^((p-1)(q-1)/4)
= 1 if p≡1(mod 4) or q≡1(mod 4)
= -1 if p≡q≡3(mod 4)
Tonelli-Shanks algorithm:
Finds x: x² ≡ a (mod p) when (a/p) = 1
Jacobi symbol (a/n):
Generalization to odd composite n
Product of Legendre symbols over prime factors
(a/n) = 1 does NOT imply a is QR mod n
Diophantine Equations
Linear Diophantine: ax + by = c
Solution exists ↔ gcd(a,b) | c
If (x₀,y₀) is one solution: x=x₀+bt/d, y=y₀-at/d (t∈ℤ)
Pythagorean triples: x² + y² = z²
Primitive triples (gcd=1): x=m²-n², y=2mn, z=m²+n²
m > n > 0, gcd(m,n)=1, m-n odd
All: (3,4,5), (5,12,13), (8,15,17), (7,24,25), ...
Pell's equation: x² - Dy² = 1 (D not perfect square)
Always has infinitely many solutions
Fundamental solution (x₁,y₁): smallest positive solution
Recurrence: xₙ + yₙ√D = (x₁+y₁√D)ⁿ
Found via continued fraction expansion of √D
Sum of squares:
n = sum of 2 squares ↔ all prime factors p≡3(mod 4) appear even power
n = sum of 4 squares: always (Lagrange's four-square theorem)
Fermat two-square theorem: p = a²+b² ↔ p=2 or p≡1(mod 4)
Fermat's Last Theorem:
xⁿ + yⁿ = zⁿ has no positive integer solutions for n ≥ 3
Proven by Andrew Wiles, 1995 (using elliptic curves + modular forms)
Catalan's conjecture (Mihailescu 2002):
xᵃ - yᵇ = 1 with a,b,x,y>1 → only solution: 3² - 2³ = 1
Analytic Number Theory
Riemann zeta function:
ζ(s) = Σₙ₌₁^∞ 1/nˢ for Re(s) > 1
Euler product: ζ(s) = Π_p 1/(1-p⁻ˢ) (over all primes)
Analytic continuation to all s ≠ 1
Trivial zeros: s = -2,-4,-6,...
Riemann hypothesis: all non-trivial zeros have Re(s) = 1/2
Prime Number Theorem:
π(x) ~ x/ln(x) as x→∞
π(x) = number of primes ≤ x
Stronger: π(x) = Li(x) + O(x·exp(-c√ln x))
Li(x) = ∫₂ˣ dt/ln(t) (logarithmic integral)
Proved independently by Hadamard and de la Vallée Poussin, 1896
Dirichlet L-functions:
L(s,χ) = Σ χ(n)/nˢ for Dirichlet character χ
Dirichlet's theorem uses L(1,χ) ≠ 0 for χ ≠ χ₀
Arithmetic progressions:
π(x;a,d) = #{p≤x: p≡a(mod d)} ~ π(x)/φ(d) (PNT for progressions)
Bertrand's postulate (proved): for n≥1, prime p with n<p≤2n
Cryptography Applications
def number_theory_crypto():
return {
'RSA': {
'key_generation': [
'Choose large primes p, q',
'n = pq, φ(n) = (p-1)(q-1)',
'Choose e: gcd(e,φ(n))=1, typically e=65537',
'Find d: ed ≡ 1 (mod φ(n)) via extended Euclidean'
],
'encryption': 'c = mᵉ (mod n)',
'decryption': 'm = cᵈ (mod n)',
'security': 'Based on hardness of factoring n',
'theorem': 'm^(ed) ≡ m (mod n) by Euler\'s theorem'
},
'Diffie-Hellman': {
'setup': 'Public: large prime p, generator g',
'Alice': 'Choose a, send A = gᵃ (mod p)',
'Bob': 'Choose b, send B = gᵇ (mod p)',
'shared key': 'K = Bᵃ = Aᵇ = g^(ab) (mod p)',
'security': 'Based on hardness of discrete log problem'
},
'Primality testing (Miller-Rabin)': {
'input': 'n odd, write n-1 = 2ˢ·d',
'step': 'Choose random a, compute a^d mod n',
'check': 'If aᵈ≢1 and a^(2ʲd)≢-1 for all j: n composite',
'iterations': 'k rounds: P(false composite) ≤ 4⁻ᵏ',
'deterministic':'For n<3.3·10²⁴, test a=2,3,5,7,11,13,17,19,23,29,31,37'
},
'Baby-step Giant-step (discrete log)': {
'problem': 'Find x: gˣ ≡ h (mod p)',
'baby_step': 'Compute gʲ for j=0,...,m-1 (m=⌈√p⌉)',
'giant_step': 'Compute h·(g^(-m))^i for i=0,...,m-1',
'match': 'When values match: x = im+j',
'complexity': 'O(√p) time and space'
},
'Elliptic Curve Cryptography': {
'curve': 'y² = x³ + ax + b (mod p)',
'group': 'Points form abelian group under chord-tangent law',
'ECDLP': 'Given P, Q=kP, find k (harder than DLP)',
'key sizes': '256-bit ECC ≈ 3072-bit RSA security',
'applications': 'TLS, Bitcoin (secp256k1), signal protocol'
}
}
Important Theorems Summary
Fermat's Little Theorem: aᵖ⁻¹ ≡ 1 (mod p) for p prime, p∤a
Euler's Theorem: a^φ(n) ≡ 1 (mod n) for gcd(a,n)=1
Wilson's Theorem: (p-1)! ≡ -1 (mod p) ↔ p prime
CRT: System of coprime moduli has unique solution mod product
Quadratic Reciprocity: (p/q)(q/p) = (-1)^((p-1)(q-1)/4)
Four Square Theorem: every positive integer is sum of four squares
FLT: xⁿ+yⁿ=zⁿ no positive integer solutions for n≥3 (Wiles 1995)
PNT: π(x) ~ x/ln(x)
Dirichlet: infinitely many primes in arithmetic progression gcd(a,d)=1
Euclid: infinitely many primes
Bezout: gcd(a,b)=d ↔ ∃x,y: ax+by=d
FTA: unique prime factorization
Common Pitfalls
| Pitfall | Fix |
|---|
| (a/p)=1 means a is QR | Jacobi symbol (a/n)=1 does NOT mean a is QR mod n |
| Fermat pseudoprimes | a^(p-1)≡1 doesn't prove p is prime; use Miller-Rabin |
| CRT requires coprime moduli | Must verify gcd(nᵢ,nⱼ)=1 for all pairs |
| φ(mn) = φ(m)φ(n) always | Only when gcd(m,n)=1 |
| Modular inverse always exists | a⁻¹ mod n exists ↔ gcd(a,n)=1 |
| Large exponent arithmetic | Always reduce mod n at each step (don't compute aᵇ first) |
Related Skills
- abstract-algebra-expert: Groups, rings, fields (algebraic number theory)
- cryptography-expert: Applied number theory
- discrete-mathematics-expert: Combinatorial number theory
- calculus-expert: Analytic number theory foundations
- computer-science-algorithms: Number theoretic algorithms