However, the main idea is pretty much the same. They both prove that the result is a common divisor first and then show that it is the greatest among all the common divisors. Note: Usually either x or y will be negative since a, b and gcd a, b are positive and both a and b are usually greater than gcd a, b. Input: Two non-negative integers a and b. This linear equation is going to be very complicated with all these notations, so it is much easier to understand with an example:.
Solve for using the second to last equation and we get:. Because by Euclidean algorithm,. Now let's solve for in the same way:. Substitute Eq. Now you can see gcd a, b is expressed by a linear combination of and. If we continue this process by using the previous equations from the list above, we could get a linear combination of and with representing and representing.
If we keep going like this till we hit the first equation, we can express gcd a, b as a linear combination of a and b, which is what we intend to do. How efficient could Euclidean algorithm be? Is it always perfect? Does Euclidean algorithm have shortcomings? Thus, the Euclidean algorithm is linear-time in the number of digits in b. We can tell from the equations that the number of steps is with n being the same n as in the division equations.
So we want to prove that where k is the number of digits of b. A theorem about the lower bound of Fibonacci numbers states that for all integers , it is true that where the sum of the Golden Ratio and 1.
Because b has k digits,. Therefore, the number of steps required by Euclidean algorithm for gcd a,b is no more than five times the number of digits of b. The Euclidean algorithm is an ancient but good and simple algorithm to find the gcd of two nonnegative integers; it is well designed both theoretically and practically.
Due to its simplicity, it is widely applied in many industries today. However, when dealing with really big integers prime numbers over 64 digits in particular , finding the right quotients using the Euclidean algorithm adds to the time of computation for modern computers. Stein's algorithm also known as the binary GCD algorithm is also an algorithm to compute the gcd of two nonnegative integers brought forward by J.
Stein in This alternative is made to enhance the efficiency of the Euclidean algorithm, because it replaces complicated division and multiplication in Euclidean algorithm with addition, subtraction and shifts, which make it easier for the CPU to compute large integers. Based on the three conclusions, Stein's algorithm is described as the following.
Note that the inner computation below is actually the same as the three conclusions. We just restate the three conclusions in an "algorithm form.
Input: any two distinctive positive integers with ;. Steiner's algorithm is designed for large numbers, but we only provide an example with small numbers for convenience.
Now you may have a better understanding of the efficiency of Stein's algorithm, which substitutes divisions with faster operations by exploiting the binary representation that real computers use nowadays.
The Euclidean algorithm is a fundamental algorithm for other mathematical theories and various subjects in different areas. Please see Application of the Euclidean Algorithm to learn more about the Euclidean algorithm. Euclid's Algorithm. Extended Euclidean Algorithm. Euclidean Algorithm. Volume 2, Second Edition. London: Cambridge University Press. Euclidean algorithm. Binary GCD Algorithm. Lame's Theorem. Induction and Recursion. From Math Images. Jump to: navigation , search. The Description of Euclidean Algorithm Mathematical definitions and their abbreviations ' [ Example: 3 6 ; 4 16 gcd means the greatest common divisor, also called the greatest common factor gcf , the highest common factor hcf , and the greatest common measure gcm.
Keep those abbreviations in mind; you will see them a lot later. Precondition The Euclidean Algorithm is based on the following theorem: Theorem: where and.
Proof: Since , could be denoted as with. Then the remainder. Assume is a common divisor of and , thus , or we could write them as Because , , so we know. Therefore is also a common divisor of. Hence, the common divisors of and are the same. In other words, and have the same common divisors, and so they have the same greatest common divisor.
If r 0, replace a by b and replace b by r. Go back to the previous step. The algorithm process is like this To sum up, is the gcd of a and b. Example An example will make the Euclidean algorithm clearer. Couldn't be 1 because the remainder has to be smaller than then quotient Couldn't be 3 otherwise it is greater than So it turns out to be 2 and the remainder is Use it as the quotient for this second equation.
By analog, find the coefficient for 40 and the remainder. Here's an applet for you to play around with finding the gcd by using the Euclidean algorithm. Recall that Eq. Proving That It Is The Greatest Second, we need to show that is the greatest among all the common divisors of a and b.
Thus, , and substitute dm for a and dn for b. Thus, , and. Euclid's Proof Now let's look at Euclid's proof. In fact even in recent times there are useful twists on the algorithm that are discovered, e. The path seems a bit deep, but here's a start, from wikipedia. The extended Euclidean algorithm was published by the English mathematician Nicholas Saunderson,[38] who attributed it to Roger Cotes as a method for computing continued fractions efficiently.
Ref 38 is Saunderson, Nicholas The Elements of Algebra in Ten Books. PDF at this archive. Ref 39 is Tattersall, J. Elementary Number Theory in Nine Chapters. Cambridge: Cambridge University Press. ISBN Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Who extended the Euclidean algorithm to derive the Bezout identity?
Ask Question. Asked 3 years, 5 months ago. Active 10 months ago. Viewed times. Who came up with the extended Euclidean algorithm? Improve this question. Conifold 62k 5 5 gold badges silver badges bronze badges. Leaky Nun Leaky Nun 1 1 silver badge 4 4 bronze badges. For a homework assignment, I derived Bezout's identity in "math camp" the Ross Mathematics Program many years ago by looking at the set of linear combinations of the two given values.
This is an existence proof, not computational and not using the Euclidean algorithm. Could you rewrite the title to this question? Further, it is not only an existence proof since it has an immediate constructive extension that leads to one form of the extended Euclidean algorithm. Add a comment. Active Oldest Votes. Improve this answer. Bill Dubuque Bill Dubuque 1 1 silver badge 7 7 bronze badges.
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