212 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

If » = 2, the same formula for x, namely
x=qgmtt—m*—n41,

givesx = — 8¢ — 5

Here the smallest positiv value of x is obtaind by putting
g = — 1, for which x = 3.

The other formula derived for x, namely

x=rmt 4+ (n—1)(m" — 1),

gives for this case x = — 87 4 3, the smallest positiv value
of x corresponding to » = 0, and

3'1=—'47‘+1
:\'2=—27’+0
g=— 1740

The monkeys get 2 coconuts
the 1st man gets — 57 + 1 *
Sl e =340 =
If » = 0, the monkeys get 2 coconuts
““ 1st man gets 1 coconut
S O ety
whose sum is 3.

Of course this solution is rather visionary, the second man
on waking would hardly have given the only remaining coconut
away; and, if they had gatherd only three, they would hardly
have waited till morning to divide them. Again, if the num-
ber gatherd was 3, they would have found it hard to divide
the coconuts equally without cracking one of them.

If » = — 1, the monkeys get 2 coconuts, the 1st man 6, the
2d man 3, whose sum is 11, the value of x when r = — 1.