An alternative GCD algorithm

The greatest common divisor ( GCD ) of two integers a and b is usually calculated with the ( well-known ) Euclidean Algorithm. There is however an alternative algorithm which is based on an entirely different idea. Let's illustrate this idea with an example. Let a, b be integers, a >:b and (a, b) = GCD(a,b). Then the following rules can be applied recursively until a=b=GCD(a,b):
If ( a=even AND b=even) then GCD(a,b)=2*GCD(a/2,b/2)
If ( a=odd AND b=even) then GCD(a,b)=GCD(a,b/2)
If ( a=even AND b=odd) then GCD(a,b)=GCD(a/2,b)
If ( a=odd AND b=odd) then GCD(a,b)=GCD(a-b,b)

Example
(36, 27) = (27, 36/2)
(27, 18) = (27, 18/2)
(27, 9) = (27-9, 9)
(18, 9) = (18/2, 9)
(9, 9) Halt.
GCD(36,27)=9.

Compare using the Euclidean Algorithm
36 = 1 * 27 + 9
27 = 3 * 9 + 0 Halt.
GCD(36,27)=9.

However, this doesn't mean that the Euclidean Algorithm is always faster.

Calculating squares

Try ( without a calculator )

21^2 ?

37^2 ?

If you need a calculator to calculate simple squares then you may need the following simple rule.

21^2 = 441

37^2 = 1369.

Or using ( x - k ) * ( x + k ) + k^2 = x^2

21^2 = 20 * 22 + 1^2 = 440 + 1 = 441

37^2 = 34 * 40 + 3^2 = 1200 + 160 + 9 = 1369.

Cubic numbers

Create a triangle from the sequence of odd numbers s[n]=2n-1 by writing s[1] on the first row, s[2] and s[3] on the second row, the next three numbers from the sequence on the third row, ... the next k numbers on the k-th row. For example:

The vertical column contains the sums by row of the numbers in the triangle. It is easy to see that this column contains the cubic numbers.

Primes

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. (Leonard Euler)

Roman numerals

Just noticed that the BBC uses Roman numerals for copyright dating. (c) MMVIII.

I - 1
V - 5
X - 10
L - 50
C - 100
D - 500
M - 1000

1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX

My goal

I decided that I should have a goal related to my math hobby. I don't know if that is good or bad. It is my goal to get this degree in mathematics. It has been in the back of my mind for a while. It's time to come out of the closet. Having a secret goal is a sort of fear of failure I guess. Working towards a goal is more fun than having achieved a goal because once a goal has been achieved new goals turn up. Talking ( writing ) about what I am doing is the purpose of this blog anyway. ( To be continued. )