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  2. Fundamental theorem of arithmetic - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of...

    The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, = =). This theorem is one of the main reasons why 1 is not considered a prime number : if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2 ⋅ 1 = 2 ⋅ 1 ⋅ 1 ...

  3. Prime number - Wikipedia

    en.wikipedia.org/wiki/Prime_number

    The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case. [119]

  4. Integer factorization - Wikipedia

    en.wikipedia.org/wiki/Integer_factorization

    Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...

  5. Prime number theorem - Wikipedia

    en.wikipedia.org/wiki/Prime_number_theorem

    The prime number theorem is obtained there in an equivalent form that the Cesàro sum of the values of the Liouville function is zero. The Liouville function is () where () is the number of prime factors, with multiplicity, of the integer .

  6. Table of prime factors - Wikipedia

    en.wikipedia.org/wiki/Table_of_prime_factors

    lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.

  7. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    The integers and the polynomials over a field share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a unit, ±1 in the case of integers) and a product of irreducible elements (prime numbers, in the case of integers), and this factorization is unique up to rearranging ...

  8. List of number theory topics - Wikipedia

    en.wikipedia.org/wiki/List_of_number_theory_topics

    Prime number, prime power. Bonse's inequality; Prime factor. Table of prime factors; Formula for primes; Factorization. RSA number; Fundamental theorem of arithmetic; Square-free. Square-free integer; Square-free polynomial; Square number; Power of two; Integer-valued polynomial

  9. Euclid's lemma - Wikipedia

    en.wikipedia.org/wiki/Euclid's_lemma

    The two first subsections, are proofs of the generalized version of Euclid's lemma, namely that: if n divides ab and is coprime with a then it divides b. The original Euclid's lemma follows immediately, since, if n is prime then it divides a or does not divide a in which case it is coprime with a so per the generalized version it divides b.