Euclid’s theorem; p w </math>
Saturday, December 29th, 2007Euclid’s theorem is generally a reference to the theorem (often credited to Euclid) which demonstrates the existence of an infinite number of prime numbers.
Assume <math>p</math> is the greatest prime number. Let <math>P=\{2,3,5,…,p\}</math> be the set of all prime numbers less than or equal to <math>p</math>, i.e. the set of all primes. Let <math>p’=2\cdot 3\cdot 5\cdot</math> <math> \cdots </math> <math>\cdot p</math> where <math>2,3,5,…,p \in P</math>. It is clear <math>p’</math> is not prime, since every element of <math>P</math> is a factor of <math>p’</math>. Let <math>q=p’+1</math>. Clearly every element of <math>P</math> divides <math>q</math> with a remainder of 1. Thus, <math>q</math> is prime or there exists a prime not in <math>P</math> which divides <math>q</math>. If <math>q</math> is prime, this would contradict our assumption that <math>p</math> is the greatest prime since <math>q>p</math>. Therefore, there must exist a prime that divides <math>q</math> which is not an element of <math>P</math>. But this would contradict our assumption that <math>P</math> is the set of all primes. Contradiction.
Today, Euclid’s result has been strengthened in a number of directions, including Dirichlet’s theorem and the prime number theorem. However, these theorems are significantly more difficult to prove.
External links
- Euclid’s Theorem at Mathworld
- energy effificient homes & utility bills - "Awnings Can Also energy efficient homes & utility bills - "Reducing fuel bills- saving the planet!" - By : Phil Ray · energy efficient homes & utility bills - "Energy Tips
- Reduce your Utility Bills? Reduce your Utility Bills? Compare Gas and Electricity Prices. Uswitch is one of the UK's leading providers of impartial advice on energy suppliers and
- HOME :: spokesmanreview.com Insulating hot-water pipes will reduce heat loss and your utility bill. And you won't have to wait as long for hot water to arrive at the tap.
- Energy Cost Management, Energy Saving Analysis - Utility Rate Reduce utility costs and receive utility bill refunds! UtiliSave will find errors and overcharges in your electric, gas, water, sewer steam, oil,
- Cheaper utility bills - Electricity for Business So how can businesses reduce their utility bills? Since the energy market was regulated, new players have entered the market, and some of them are at last
- Reduce Utility Bills » Gas The benefits to doing this not only reduce utility bills, but also is good for the environment because of the reduced energy use.
- Frugal Friday: Reduce Utility Bills Reducing monthly bills is just one more way to save money on your household budget.