*mean*of two numbers

*a*and

*c*as their average

*b*= 1/2(

*a*+

*c*). Another way to think of

*b*is that it is the number that makes

*a*,

*b*,

*c*an arithmetic sequence, so

*b*is more properly called the

*arithmetic mean*of

*a*and

*c*. Similarly, we can find the geometric mean of two numbers,

*a*and

*c*. This time,

*b*will be the number that makes

*a*,

*b*,

*c*into a geometric sequence. The geometric mean of two numbers

*a*and

*c*is given by

*b*= sqrt (

*a**

*c*).

Here is a construction for the geometric mean of two numbers that uses a parabola in an interesting way:

- Graph the parabola
*y*=*x*^2. - Plot the points
*a*and*c*on the*y*axis: let*A*be (0,*a*) and let*C*be (0,*c*). - Draw a line through
*A*parallel to the x axis - the point where it touches the parabola's positive arm is*A*'. Do the same for*C*- the point where it touches the parabola's negative arm is*C*'. - Draw the line through
*A*'*C*'. The point where this line crosses the*y*axis is*B*= (0,*b*), where*b*is the geometric mean of*a*,*c*.

This construction of the geometric mean can be used to create a geometric prime sieve. Usually, we visualize the sieve of Eratosthenes on a number chart. Starting with 2, w cross out all the multiples of 2 (except for 2 itself): these numbers are clearly not prime. Then we continue with the first number that we did not cross out, 3, and cross out all its multiples (except for 3 itself)... if we continued indefinitely we will cross out all composites, leaving only primes behind (we have woven a sieve that only lets the primes fall through). Practically, if we are hunting for primes less than

*n*, we only need to go as far as crossing out multiples of primes up to sqrt(

*n*).

Here is how we can construct a similar prime sieve using a parabola - the geometric mean construction can help you see why we are able to hit all the composites.

1. Graph the parabola

*y*=

*x*^2.

2. Plot the points on the parabola obtained from

*x*= -2, -3, -4, ...

3. Plot

*x*= 2 on the parabola. Draw lines from this point on the parabola to each point drawn in step 2. The

*y*-intercepts of these lines are the multiples of 2.

4. Plot

*x*= 3 on the parabola. Draw lines from this point on the parabola to each point drawn in step 2. The

*y*-intercepts of these lines are the multiples of 3.

5. Continue the process for

*x*values that are not among the multiples found in previous steps.The numbers on the

*y*axis, greater than 1 that are not touched by the constructed lines are the primes left behind by the sieve.

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