# Euclidean Algorithm terminates in at most 7x the digits of b

## Homework Equations

I'm not sure if this is relevant:
##r_2 \leq \frac{1}{2}r_1## ... ##r_n \leq (\frac{1}{2})^nr_1##

## The Attempt at a Solution

i have shown that ##r_{i+2} < r_i## by showing the ##r_{i+2} - r_i## is negative, but how do I show that the number of steps is at most ##2 \log_{2}b##

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andrewkirk
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i have shown that ##r_{i+2} < r_i##
That is not what the question requires though. It requires to show that ##r_{i+2} < \frac12 r_i##. Have you shown that?
how do I show that the number of steps is at most ##2 \log_{2}b##
Once you have shown ##r_{i+2} < \frac12 r_i## you can work out how many steps it takes to reduce the remainder to 1 or less, as the algorithm terminates when the remainder reaches 1 or 0. Use the fact that the remainder halves every two steps and that initially it is no greater than ##b##..

That is not what the question requires though. It requires to show that ##r_{i+2} < \frac12 r_i##. Have you shown that?
Once you have shown ##r_{i+2} < \frac12 r_i## you can work out how many steps it takes to reduce the remainder to 1 or less, as the algorithm terminates when the remainder reaches 1 or 0. Use the fact that the remainder halves every two steps and that initially it is no greater than ##b##..
Sorry I meant ##2r_{i+2} < r_i##

Use the fact that the remainder halves every two steps and that initially it is no greater than bbb..
I am not sure if I am doing this completely right or wrong.

proof of ##r_{i+2} < \frac{1}{2}r_i:##

Let ##a - bq_1 = r_1## s.t. ##b=r_0##

##(case 1):##
If ##b=r_0< \frac{1}{2}a## then ##q_1 \geq 2## so that ##r_1 < r_0##.
Furthermore, in ##r_0 - r_1q_2 = r_2##, since ##r_1 \leq \frac{1}{2}r_0## then it must be that ##q_2 \geq 2## so that ##r_2 < r_1##.
Therefore, ##r_2 < r_1 < \frac{1}{2}r_0## or ##r_{i+2}<r_{i+1}< \frac{1}{2}r_i## or simply ##r_{i+2}<\frac{1}{2}r_i##.

##(case 2):##
If ##b=r_0> \frac{1}{2}a## then ## a- r_0q_1 = r_1## such that ##r_1 < \frac{1}{2}a## and ##r_1 < r_0## and ##q_1=1##

##(subcase 2.1):## If ##r_1 > \frac{1}{2}r_0## then definitely, ##r_2 < \frac{1}{2}r_0## or ##r_{i+2}<\frac{1}{2}r_i##

##(subcase 2.2)## If ##r_1 < \frac{1}{2}r_0## then ##q_2 \geq 2## so that ##r_2 < r_1##.
Therefore, ##r_2<r_1< \frac{1}{2}r_0## or ##r_{i+2}<r_{i+1}< \frac{1}{2}r_i##

From here, I am struggling to apply logarithms to show the final step. Can you give me hints?

SammyS
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You may get more response if you show the image rather than just giving the link. I'll do that much for you.

The image for the link you posted: Notice that if ##\ b\ ,\ ## which is also ##\ r_0\ ,\ ## has k decimal digits, then b < 10k .

Last edited:
• Terrell
andrewkirk
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Can you give me hints?
The hint is the last paragraph of post #2.

SammyS
Staff Emeritus
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Can you give me hints?
Another hint:

What upper bound on rn will ensure that the algorithm terminates in at most n steps.

I figured that the inequality ##b(\frac{1}{2})^\frac{n}{2} < 1## then taking the log of both sides should be enough...?

What upper bound on rn will ensure that the algorithm terminates in at most n steps.
##r_n \leq 1##

Notice that if b , b , \ b\ ,\ which is also r0 , r0 , \ r_0\ ,\ has k decimal digits, then b < 10k .
yes i do notice, but can this be helpful to particularly show that it takes less than 7 times the no. of digits of b?

SammyS
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I figured that the inequality ##b(\frac{1}{2})^\frac{n}{2} < 1## then taking the log of both sides should be enough...?
It's probably enough for ##\displaystyle b(\frac{1}{2})^\frac{n}{2} < 2\ .\ ## Right.

Combine this with b < 10k .

So if ##\displaystyle 10^k (\frac{1}{2})^\frac{n}{2} < 2\ ,\ ## then ...

• Terrell
Combine this with b < 10k
##2\log_2(b)<n## then by letting ##b = 10^k## we get ##2k\log_2(10)<n##. Also ##2\log_2(b)=6.6438...## therefore it can be at most 7 times the number of digits of ##b##. Thank you Sammy!