Miscellaneous Problems 239
the points of the Mandelbrot set on the complex plane. There is, of course,
the problem of deciding whether or not the terms of a sequence which
initially does not wander too far from 0 will eventually remain bounded.
Is there any way of deciding how many terms to compute before c can be
put in the Mandelbrot set or not?
Alternatively, one can fix a value of c and begin the iteration with dif-
ferent values of zi. Let P, be the set of zi for which the sequence remains
within some disc of finite radius. What does PO look like? PI? Try to plot
points in PC for various values of c.
E.68. Sums of Two Squares. One of the most celebrated results of
Leonard Euler (1707-1783) is that every prime of the form 4n + 1 can be
written as the sum of two squares. From a certain identity (which you can
derive for yourself by examining the squares of the absolute values of a + bi,
c+ di and the product (u + bi)(c + di)), i t. can be shown that every number
which is the product of sums of two squares is itself a sum of two squares.
Moreover, every product of factors of the form 2”, p” and q2’“, where U,
v, w are nonnegative integers and p, q are primes with p f 1 (mod 4) and
q E 3 (mod 4), can be expressed as the sum of two squares.
There is an algorithm which determines the representation of a prime
4n + 1 as the sum of two squares. Define a transformation T on number
triples as follows:
Ttx, Y, %I=
1(X-Y- z,y,2y+z) ifx>y+z
(y+z-x,2,2x-z) ifx<y+z.Verify that the quantity 4xy + z2 is left invariant by this transformation.
Start with (n, 1,1) and apply T repeatedly. For example, when n = 8,9,
we obtain the chains
(8,Ll) - (6,193) - (2,1,5) - (4,2,-l)- WJ) - PJJ) - (4,2,1)
- (GW - (6,1,-3) - (8,1, -1)
- (B,l,l) - ...
(9,Ll) - (7,I,3) - (3,135) - (3,3,1) - (1,3,5) - (7,1,-3) - (9,1,-l)
- (9,1,1) - ***
If, somewhere in the chain, we find the triple (r,r,s), show that 472 + 1 =
(2~)~ + s2. For example, the chain (9,1,1) contains (3,3,1) and we find
that 37 = 62 + 12.
If the number 4n+l is expressible as the sum of two squares, will the chain
beginning with (n, 1,l) always contain the form (r, r,s)? Does repeated
application of T to (n, 1,1) always return to (n, 1, l)? If so, what can be
said about the length and symmetry of the cycle? If not, what are the
exceptional values of n?