Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic states that {n: n is an element of N > 1} (the set of natural numbers, or positive integers, except the number 1) can be represented uniquely apart from rearrangement as the product of one or more prime numbers (a positive integer that's divisible only by 1 and itself). This theorem is also called the unique factorization theorem and is a corollary to Euclid's first theorem, or Euclid's principle, which states that if p is a prime number and p/ab is given (a does not equal 0; b does not equal 0), then p is divisible by a or p is divisible by b.

Proof: First prove that every integer n > 1 can be written as a product of primes by using inductive reasoning. Let n = 2. Since 2 is prime, n is a product of primes. Suppose n > 2, and the above proposition is true for N < n. If n is prime, then n is a product of primes. If n is composite, then n = ab, where a < n and b < n. Therefore, a and b are products of primes. Hence, n = ab is also a product of primes. Since that has been established, we can now prove that such a product is unique (except for order). Suppose n = p sub1 * p sub2 * ... * p subk = q sub1 * q sub2 * ... * q subr, where the p's and q's are primes. If so, then p sub1 is divisible by (q sub1 * ... * q subr) by Euclid's first theorem. What is the relationship between p sub1 and one of the q's? If the r in q subr equals 1, then p sub1 = q sub1 since the only divisors of q are + or - 1 and + or - q and p > 1, making p = q. What about the other factors in the divisor? If p does not divide q, then the greatest common denominator of p and q is 1 since the only divisors of p are + or - 1 and + or - p. Thus there are integers m and n so that 1 = am + bn. Multiplying by q subr yieds q subr = amq subr + bnq subr. Since we are saying that p is divisible by q, let's say the q sub1 * q subr = cp. Then q subr = amq subr + bnq subr = amq subr + bcm = m(aq subr + bc). Therefore, p is divisible by q sub1 of q sub2 * ... * q subr. If p sub1 is divisible by q sub1, then p sub1 = q sub 1. If this does not work the first time, then repeat the argument until you find an equality. Therefore, one of the p's must equal one of the q's. In any case, rearrange the q's so that p sub1 = q sub1, then p sub1 * p sub2 * ... * p subk= p sub1 * q sub2 * ... * q subr and p sub2 * ... * p subk = q sub2 * ... * q subr, and so on. By the same argument, we can rearrange the remaining q's so that p sub2 = q sub2. Thus n can be expressed uniquely as a product of primes regardless of order, making the fundamental theorem of arithmetic true.

by some punk kid September 6, 2005

arithmatic

You're a dumbass who can't spell arithmetic.

You meant to do a search for arithmetic, not "arithmatic", you moron.

by Cixelsid September 19, 2009

chinese arithmetic

A descriptive term used to describe an extreme erection.

When Katrina was sucking my dick and it got harder than fucking chinese arithmetic.

by Carl Pickens March 23, 2007

Arithmetard

One who is a complete retard when it comes to even the simplest math. Especially applicable when that person should know math for their job.

I am such an arithmetard! I'm a tenured professor of theoretical physics, but can I calculate a simple tip? NO!

by JoeyIsConfused January 14, 2009

arithmedick

Arithmedick- A math equation used to identify if a female is a HOE (Horizontally Orientated Entertainer) the equation is as follows.

Number of Males said female has slept with X 6 (For average number of times each male penetrated Slot-C)= Z you then take Z and place a platform at that height and if a fall from the platform is fatal she is a HOE. Example: Debbie slept with 15 guys her freshman year so 15guys X 6 = 90 or 90feet a fall from 90 feet is fatal concluding Debbie is a Horizontally Orientated Expert.

Tammy slept with 6 guys so by the arithmedick equation 6X6 equals 36 so if she fell from a platform of 36 feet it's not fatal so Tammy isn't a hoe....yet.

by JohnnyBadGood July 18, 2016

Ardithakur

Brown Indian, small, a nerd and has a red dot on his five head. His girlfriend Natalie can’t even have natural hair colour so he has to buy hair dye all the time and he’s very racist because he’s jealous that all of us have better things then him.

Ardithakur: rajabaja son calling from India

Us: bye
Ardi: no no raja man wait arr we offer $60 a month and no internet

Us: please do not call me again

Ardi: please I am forced to play cricket help me get some rupees and I love nethme
Us: sorry I can’t give you rupees and I’m not going to help you with that

by Ardicurry ethanfatty ashdaddy June 2, 2020

arithmaticionopiakeripeonlasioususackersonaperithmeonrasissleoepeonjarrithonkasapotusiousoerithionosouras

arith-matic-eon-opia-kerip-eon-las-e-o-sus-ack-er-so-na-per-ith-me-on-ra-siss-l-o-e-pee-on-jar-ith-on-ka-sa-po-tus-e-us-o-er-eth-e-on-o-saurus: the only reason this word exists is to be the longest and hardest to pronounce

arithmaticionopiakeripeonlasioususackersonaperithmeonrasissleoepeonjarrithonkasapotusiousoerithionosouras is the longest word

by judok September 20, 2020