المبرهنة الأساسية في الحساب

المبرهنة الأساسية في الحساب Fundamental theorem of arithmetic أو ما يعرف بمبرهنة التحليل إلى جداء أعداد أولية هي مبرهنة رياضية تنص على أن كل عدد صحيح طبيعي غير منعدم يمكن كتابته على شكل جداء أعداد أولية, و هذه الكتابة وحيدة. ومثال ذلك:

$6936=2^{3}\times 3\times 17^{2}$  أو  $1200=2^{4}\times 3\times 5^{2}$

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 التطبيقات

 التمثيل القانوني لعدد صحيح موجب

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

$n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}=\prod _{i=1}^{k}p_{i}^{\alpha _{i}}$

where p₁ < p₂ < ... < p_k are primes and the α_i are positive integers.

This representation is called the canonical representation of n, or the standard form of n.

For example 999 = 3³×37, 1000 = 2³×5³, 1001 = 7×11×13

Note that factors p⁰ = 1 may be inserted without changing the value of n (e.g. 1000 = 2³×3⁰×5³).
In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers,

$n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots =\prod p_{i}^{n_{p_{i}}},$

where a finite number of the n_i are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers.

 العمليات الحسابية

This representation is convenient for expressions like these for the product, gcd, and lcm:

$a\cdot b=2^{a_{2}+b_{2}}\,3^{a_{3}+b_{3}}\,5^{a_{5}+b_{5}}\,7^{a_{7}+b_{7}}\cdots =\prod p_{i}^{a_{p_{i}}+b_{p_{i}}},$

$\gcd(a,b)=2^{\min(a_{2},b_{2})}\,3^{\min(a_{3},b_{3})}\,5^{\min(a_{5},b_{5})}\,7^{\min(a_{7},b_{7})}\cdots =\prod p_{i}^{\min(a_{p_{i}},b_{p_{i}})},$

$\operatorname {lcm} (a,b)=2^{\max(a_{2},b_{2})}\,3^{\max(a_{3},b_{3})}\,5^{\max(a_{5},b_{5})}\,7^{\max(a_{7},b_{7})}\cdots =\prod p_{i}^{\max(a_{p_{i}},b_{p_{i}})}.$

While expressions like these are of great theoretical importance their practical use is limited by our ability to factor numbers.

 الدوال الحسابية

المقالة الرئيسية: Arithmetic function