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Diophantine equations

Integer solutions: linear equations via Bézout, Pythagorean triples, Fermat.

ax + by = c has integer solutions exactly when gcd(a, b) divides c, and the extended Euclidean algorithm finds them. Pythagorean triples are (m² − n², 2mn, m² + n²). Fermat's last theorem says xⁿ + yⁿ = zⁿ has none for n ≥ 3 — proved in 1995. Picture it: lattice points on a line; the solutions are evenly spaced. Think it: Hilbert's tenth problem showed no algorithm can decide all Diophantine equations.

Mfano wenye matokeo: gcd(12, 18)

Number-Theory · Gcd-And-Lcm

Gcd(12, 18)

12,\ 18

Hatua kwa hatua

  1. \gcd(12, 18)

    Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.

  2. 18 = 1 \times 12 + 6

    Divide 18 by 12: quotient 1, remainder 6.

  3. 12 = 2 \times 6 + 0

    Divide 12 by 6: quotient 2, remainder 0.

  4. \gcd(12, 18) = 6

    The last non-zero remainder is 6.

Lafunua jibu
\gcd(12, 18) = 6

Symbols used here

a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
a \bmod n
remainder
What is left after dividing a by n.

How to: Diophantine equations

  1. Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.
  2. Divide 18 by 12: quotient 1, remainder 6.
  3. Divide 12 by 6: quotient 2, remainder 0.
  4. The last non-zero remainder is 6.

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

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Mengi zaidi katika Number Theory

Prime factorisationPrime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesFermat's little theorem and Euler's theoremRSA: cryptography from number theory