maths.free › Number Theory › Diophantine equations

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.

工作范例 · gcd(12, 18)

Number-Theory · Gcd-And-Lcm

Gcd(12, 18)

12,\ 18

一步

  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.

发送答案
\gcd(12, 18) = 6

现在轮到你 选择问题, 输入或绘制您自己的 。 每一步, 图片, 答案隐藏到您询问 。

保持你自己的工作

免费账户在每一课上都添加注解, 记录你完成的, 在一个地方解决的, 以及您可以询问的关于此页的导师。 数学本身向每个人开放, 是否签名 。

签名上签名 登录

此处所用符号

输入完整定义、图片和其中每个字母的含义的符号。

如何: 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.

人们对人们提出的问题

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.

更多 Number Theory

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