Solution to Problem 59


Congratulations to this week's winner

Nathan Pauli

A partial solution was also received from Ray Kremer. Further correct solutions were also received from William Webb, Robert McQuaid, Steven Young, Burkart Venzke. Two incorrect solutions were submitted.

Many different solutions were submitted.  The one which follows contains ideas from various solutions as well as some ideas of mine.

Start with the cubic f(x) = ax3 + bx2 + cx + d, where the coefficents a,b,c,d are all integers.   In order for the stated conditions to hold, the roots of the first and second derivative of f(x) must be integers, and the roots of the first derivative must be distinct.  The second derivative is f ''(x) = 6ax + 2b with the root b/3a; in particular, 3a divides b.

Now consider the cubic polynomial g(x) = f (x - b/3a) which is also polynomial with integer coefficients, and whose graph is the same as that of f(x) only shifted to the right by b/3a units.  Therefore f(x) satisfies the desired conditions exactly when g(x) does.  This is handy because a little algebra with convince you that the equation for g(x) has its coefficient of x2 equal to 0.  So let's write

g(x) = Ax3 + Cx + D,
with integer coefficients A, C, D and with derivatives g'(x) = 3Ax2 + C,  g''(x) = 6Ax.  The point of inflection is at x = 0 so the third condition is satisfied; the local extrema are at Ö-(C/3A) and -Ö-(C/3A), which are distinct exactly when C is unequal to 0 and are integers only when
C = - 3An2,  for some non-zero integer n.

In particular, A and C are of opposite sign. Notice that the local extrema occur at +n and -n, so, for g(x) to have three distinct roots, g(n) and g(-n), which are the local maximum and local minimum values, must be of opposite sign; this can be guaranteed by the condition g(n)g(-n) < 0.  Working out the algebra gives

g(n)g(-n) = D2 -(2An3)2 < 0
or equivalently
D lies between -2An3 and 2An3.

The final result is that

f(x) = g(x - r) = A(x - r)3 - 3An2(x - r) + D
where
A, m, n, D are integers and D lies between -2An3 and 2An3

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Page last updated 8 April 1999.