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'''[[Euler]]'s factorization method''' is a technique for [[integer factorization|factoring]] a number by writing it as a sum of two squares in two different ways. For example the number <math>1000009</math> can be written as <math>1000^2 + 3^2</math> or as <math>972^2 + 235^2</math> and Euler's method gives the factorization <math>1000009 = 293 \cdot 3413</math>.
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The idea that two distinct representations of an odd positive integer may lead to a factorization was apparently first proposed by [[Marin Mersenne]]. However, it was not put to use extensively until Euler one hundred years later. His most celebrated use of the method that now bears his name was to factor the number <math>1000009</math>, which apparently was previously thought to be prime even though it is not a [[pseudoprime]] by any major primality test.
 
Euler's factorization method is more effective than Fermat's for integers whose factors are not close together and potentially much more efficient than trial division if one can find representations of numbers as sums of two squares reasonably easily. Euler's development ultimately permitted much more efficient factoring of numbers and, by the 1910s, the development of large factor tables going up to about ten million{{Citation needed|date=July 2013}}. The methods used to find representations of numbers as sums of two squares are essentially the same as with finding differences of squares in Fermat's factorization method.
 
The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime factor of the form 4''k''&nbsp;+&nbsp;3 occurring to an odd power in its prime factorization, as such a number can never be the sum of two squares. Even odd [[composite number]]s of the form 4''k''&nbsp;+&nbsp;1 are often the product of two primes of the form 4''k''&nbsp;+&nbsp;3 (e.g. 3053 = 43 &times; 71) and again cannot be factored by Euler's method.
 
This restricted applicability has made Euler's factorization method disfavoured for [[computer]] factoring [[algorithm]]s, since any user attempting to factor a random integer is unlikely to know whether Euler's method can actually be applied to the integer in question. It is only relatively recently that there have been attempts to develop Euler's method into computer algorithms for use on specialised numbers where it is known Euler's method can be applied.
 
==Theoretical basis==
 
The [[Brahmagupta–Fibonacci identity]] states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, given <math>n = a^2 + b^2 = c^2 + d^2</math> we find <math>n</math> as a product of sums of two squares.
 
First deduce that
 
:<math>a^2 - c^2 = d^2 - b^2</math>
 
and factor both sides to get
 
:<math>(a-c)(a+c) = (d-b)(d+b)</math> (1)
 
Now let <math>k = \operatorname{gcd}(a-c,d-b)</math> and <math>h = \operatorname{gcd}(a+c,d+b)</math> so that there exists some constants <math>l,m,l',m'</math> satisfying
 
* <math>(a-c) = kl</math>,
* <math>(d-b) = km</math>,
<math>\operatorname{gcd}(l,m) = 1</math>
* <math>(a+c) = hm'</math>,
* <math>(d+b) = hl'</math>,
<math>\operatorname{gcd}(l',m') = 1</math>
 
Substituting these into equation (1) gives
 
:<math>klhm' = kmhl'</math>
 
Canceling common factors yields
 
:<math>lm' = l'm</math>
 
Now using the fact that <math>(l,m)</math> and <math>(l',m')</math> are pairs of relatively prime numbers, we find that
 
* <math>l = l'</math>
* <math>m = m'</math>
 
So
 
* <math>(a-c) = kl</math>
* <math>(d-b) = km</math>
* <math>(a+c) = hm</math>
* <math>(d+b) = hl</math>
 
We now see that <math>m = \operatorname{gcd}(a+c,d-b)</math> and <math>l = \operatorname{gcd}(a-c,d+b)</math>
 
Applying the [[Brahmagupta–Fibonacci identity]] we get
 
:<math>(k^2 + h^2)(l^2 + m^2) = (kl - hm)^2 + (km + hl)^2 = ((a-c) - (a+c))^2 + ((d-b) + (d+b))^2 = (2c)^2 + (2d)^2 = 4n,</math>
 
:<math>(k^2 + h^2)(l^2 + m^2) = (kl + hm)^2 + (km - hl)^2 = ((a-c) + (a+c))^2 + ((d-b) - (d+b))^2 = (2a)^2 + (2b)^2 = 4n.</math>
 
As each factor is a sum of two squares, one of these must contain both even numbers: either <math>(k, h)</math> or <math>(l ,m)</math>. Without loss of generality, assume that pair <math>(k,h)</math> is even. The factorization then becomes
 
:<math>n = ((\tfrac{k}{2})^2 + (\tfrac{h}{2})^2)(l^2 + m^2). \,</math>.
 
==Worked example==
 
Since: <math>\ 1000009 = 1000^2 + 3^2 = 972^2 + 235^2</math>
 
we have from the formula above:
 
{| class="wikitable"
|-
|''a'' = 1000
|(A) ''a'' &minus; ''c'' = 28
|gcd[A,C] ''k'' = 4
|-
|''b'' = 3
|(B) ''a'' + ''c'' = 1972
|gcd[B,D] ''h'' = 34
|-
|''c'' = 972
|(C) ''d'' &minus; ''b'' = 232
|gcd[A,D] ''l'' = 7
|-
|''d'' = 235
|(D) ''d'' + ''b'' = 238
|gcd[B,C] ''m'' = 58
|}
 
Thus,
 
: <math> 1000009 = \left[\left(\frac{4}{2}\right)^2 + \left(\frac{34}{2}\right)^2\right] \cdot \left(58^2 + 7^2\right) \, </math>
:: <math>= \left(2^2 + 17^2\right) \cdot \left(58^2 + 7^2\right) \, </math>
:: <math>= (4 + 289) \cdot (3364 + 49) \, </math>
:: <math>= 293 \cdot 3413 \, </math>
 
==References==
* {{Cite book|chapter=Euler's Factorization Method|last= Ore|first= Oystein|title=Number Theory and Its History|pages=59–64| isbn= 0-486-65620-9}}
 
* {{Cite journal|last=McKee|first= James|title=Turning Euler's Factoring Method into a Factoring Algorithm|journal=Bulletin of the London Mathematical Society|year= 1996| issue =28 |volume= 4|pages= 351–355|doi=10.1112/blms/28.4.351}}
 
{{number theoretic algorithms}}
 
[[Category:Integer factorization algorithms|Euler's factorization method]]

Latest revision as of 04:35, 15 August 2014

In fact, many small enterprises do not understand the process or complain that doing an industry research is too complex or too expensive and wonder why or if it is necessary.

What is market research?

In the most basic conditions, a market research can be an examination of:

- A particular problem or opportunity in an industry.

- The n...

The term "market research" is frequently puzzling to entrepreneurs, especially for people who concentrate on a certain market or market segment.

In fact, many small enterprises don"t understand the procedure or complain that conducting an industry research is too difficult or too expensive and wonder why or when it is necessary.

What"s market research?

In one of the most simple terms, market research is an review of:

- A specific problem or opportunity in a market.

- The wants of the target market regarding the issue or opportunity.

- Some ideas for advertising a certain product or service that fills the needs of the target market.

When should you perform an industry research?

- If you are starting a small business.

- When you"re entering a new industry.

- If you are considering a new products or services.

Why should you perform market analysis?

- To reduce business risks.

- To know the possibilities and problems.

- To identify sales opportunities. In case you fancy to identify further about more information, we know about tons of resources you can investigate.

- To plan your marketing/sales strategy.

The process of conducting a market analysis could be divided in to

three parts:

Part 1 - Understanding Industry Problems

This provides you essential information about your entire industry -- the size, your competitors, the customers.

Part 2 - Identifying Industry Opportunities

This gives you more specific information about potential problems or opportunities in-the potential market, and includes information about more information, present and future developments, external facets and development about particular opponents.

Part 3 - Devel-oping Market-Driven Strategies

Here is where we enter what market research does for-you. It will help you to pinpoint opportunities to grow your company. By knowing the industry and knowing what options are available you"ll be able to develop a marketing strategy that leaves your competition in the dust!

Listed below are 10 questions that will help you obtain started:

1. What"s industry I do want to reach?

- Who are they? (Basic Demographics)

- What"s their greatest problem in terms of forex trading?

- Are their requirements being met by these products or services offered in the forex market?

2. Who is my opposition in this market?

- Are they successful in forex trading?

- Are they marketing an identical service or product?

- What is the market share of the three greatest competitors in this market?

3. Is there room for growth in the forex market?

4. For alternative viewpoints, please consider glancing at: fundable. What"s how big the forex market?

- Will there be room for growth?

- Could be the market growing? Firm? Unhealthy? Unpredictable? Declining?

5. How is my product or service different from the competition?

6. How to reach this market?

- How is my opposition currently reaching forex trading?

- Can it be the very best way?

- What are the alternative ways of reaching the forex market?

7. What"re the business models of my competition in forex trading?

- Are they successful?

- Can there be ways to do it differently or better?

8. What do customers expect from this type of service or product?

- What"re the core competencies of the service or product?

- What could make the item "new" "different" or "better" for the consumer?

9. I discovered staples fundable by searching Google. How much are clients willing to purchase this product or service?

1-0. What"s our competitive advantage in this market?

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