PHI by Joseph K. Horn   --  A Number Theory tool.
              Calculates PHI(x), Euler's totient function.

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| NOTE:  This program calls FACTOR, a program by Jurjen NE Bos, available |
| on EduCALC Goodies Disk #2.  You must be in the FACTOR directory when   |
| you upload ALLF, or it will not work.                                   |
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PHI(x) is defined as the number of integers between 1 and x that are
relatively prime to x (i.e. share no prime factors with x).

For example, since 15 is relatively prime to 1, 2, 4, 7, 8, 11, 13 and
14 (8 numbers), PHI(15) is 8.

By convention, 1 is neither a prime nor a composite, and is therefore
considered relatively prime to everything.  Thus PHI(1)=1.

PHI of anything less than 1 yields 0.

The number of primitive roots of x is exactly equal to PHI(PHI(x)). For
example, to see that 351 has 72 primitive roots, type 351 and press PHI
twice.  This means that there are 72 numbers between 1 and 351 which are
relatively prime to 351.  Anything that can be described in terms of
close-range random but long-range patterns can be modeled by primitive
root cycles.  The applications of this range from the SpiroGraph child's
toy to the mathematics of Chaos Theory.

-Joseph K. Horn-    -Peripheral Vision, Ltd.-