The order of an element x within G is the least n such that xn equals the identity element. If there is no such n then the order of x is infinite. If G is finite then the order of every x in G is finite. You have to cycle around to E eventually.
These are standard terms in group theory, and they apply to elliptic curves. For example, every elliptic group over a finite field has finite order. This was discussed in the previous section.
Let G be an elliptic curve in the complex plane. Let p, an element of G, have coordinates x,y. The tangent formula sets m = (3x2+a)/2y. then the "next" value of x is m2-2x. this is the "doubling" formula.
How does the input compare to the output? When does the point move farther from the origin? When is the "next" value of x guaranteed to have a larger absolute value?
If m is less than the square root of x, in absolute value, then m2-2x is larger. If this holds for all x beyond a certain radius r, and p lies outside this radius, then p has infinite order. Each doubling moves it farther from the origin, and this continues forever without repeating a value.
Assume |x2| is at least twice |a|, and |x3| is at least four times |b|.
Now a can contribute no more than half of x2. the numerator in the formula for m becomes 7x2/2. It can't be any larger than that in absolute value.
The denominator is twice x3+ax+b, or (x2+a)x+b. This can't be any smaller than x3/2+b, which is no smaller than x3/4.
Put this all together and m is at most 7/x. Let x be at least 4 and m2 is no larger than x.
In summary, |x|2 ≥ |2a|, |x|3 ≥ |4b|, and |x| ≥ 4. I'm sure you could find tighter bounds, but this will do.
For any given ellliptic curve, most of the values of x lie outside these bounds. In other words, all the points beyond a certain radius have infinite order in G.