Euclid's lemma

(noun)

One of the fundamental properties of prime numbers. States that if a prime divides the product of two numbers, it must divide at least one of the factors. For example since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7 = 133. It is used in the proof of the fundamental theorem of arithmetic.

Related Terms

Examples of Euclid's lemma in the following topics:

  • Scientific Applications of Quadratic Functions

    • This has been proven in many ways, among the most famous of which was devised by Euclid.
    • Euclid used this diagram to explain how the sum of the squares of the triangle's smaller sides (pink and blue) sum to equal the area of the square of the hypotenuse.

  • Technological Advancements under the Song

    • Yang Hui also provided rules for constructing combinatorial arrangements in magic squares, provided theoretical proof for Euclid's forty-third proposition about parallelograms, and was the first to use negative coefficients of "x" in quadratic equations.

  • Scientific Advancements in the Classical Period

    • The discoveries of several Greek mathematicians, including Pythagoras and Euclid, are still used in mathematical teaching today.

  • The Islamic Golden Age

    • Scientists recovered the Alexandrian mathematical, geometric, and astronomical knowledge, such as that of Euclid and Claudius Ptolemy.

  • Arts and Sciences

    • By 1200 there were reasonably accurate Latin translations of the main works of Aristotle, Euclid, Ptolemy, Archimedes, and Galen—that is, all the intellectually crucial ancient authors except Plato.