MEI Maths Item of the Month
August 2014
Cubic with Integer Points
Find a cubic equation, with three distinct roots, such that the roots and the stationary points
all lie on points with integer coordinates.
If the coefficient of x³ is equal to 1, what is the cubic with this property where the sum of the
absolute value of the roots is as small as possible?
Solution
By assuming one of the roots is at the origin the curve can be written as y  x( x  a)( x  b) ,
where a and b are positive integers:
Expanding y  x( x  a)( x  b) gives y  x 3  (a  b) x 2  abx .
dy
 3x 2  2(a  b) x  ab
dx
b  a  a 2  ab  b 2
dy
Solving
 0 gives x 
3
dx
Showing a and b are multiples of 3
b  a  a 2  ab  b 2
x
is only an integer when b  a and
3
a 2  ab  b2 are multiples of 3.
If b  a is a multiple of 3 then a and b are either both multiples of 3, both 1 more than a
multiple of 3 or both 2 more than a multiple of 3.
a 2  ab  b2 must be a multiple of 9 for
a 2  ab  b2 to be a multiple of 3.
1 of 2
v1.0 19/12/2014
© MEI
MEI Maths Item of the Month
a and b are both multiples of 3:
(3m) 2  (3m)(3n)  (3n) 2  9m 2  9mn  9n 2
 9(m2  mn  n 2 )
a and b are both 1 more than a multiple of 3:
(3m  1) 2  (3m  1)(3n  1)  (3n  1) 2  9m 2  9m  9mn  9n 2  9n  3
 9(m2  m  mn  n 2  n)  3
a and b are both 2 more than a multiple of 3:
(3m  2) 2  (3m  2)(3n  2)  (3n  2) 2  9m 2  18m  9mn  9n 2  18n  12
 9(m2  2m  mn  n 2  2n  1)  3
Therefore a and b must both multiples of 3 for a 2  ab  b 2 to be a multiple of 9.
Systematically trying multiples of 3 for a and b
ba
a
3
3
3
6
3
6
3
6
9
3
6
9
3
6
9
12
b
3
6
9
6
12
9
15
12
9
18
15
12
21
18
15
12
a  ab  b
2
2
3
1.732051
3.645751
5.605551
3.464102
7.582576
5.358899
9.567764
7.291503
5.196152
11.55744
9.244998
7.082763
13.54983
11.2111
9
6.928203
ba
a  ab  b
2
2
3
-1.73205
-1.64575
-1.60555
-3.4641
-1.58258
-3.3589
-1.56776
-3.2915
-5.19615
-1.55744
-3.245
-5.08276
-1.54983
-3.2111
-5
-6.9282
The first solution is a  9, b  15 (or a  15, b  9 ). This gives:
y  x( x  9)( x  15)
 x 3  6 x 2  135 x
Roots: (9, 0), (0, 0), (15, 0) .
dy
 3x 2  12 x  135
dx
 3( x  9)( x  5)
Stationary points: (5, 400), (9, 972) .
2 of 2
v1.0 19/12/2014
© MEI