2009/006) integers x = a3+ b3+ c3-3abc for some integers a,b,c. prove that if x,y € S then xy €S.

Let S be set of integers x such that x = a3+ b3+ c3-3abc for some integers a,b,c. prove that if x,y € S then xy €S.


Proof:

we know
a^3+ b^3+ c^3-3abc = (a+bw+cw^2)(a+bw^2+cw) where w = cube root of -1

let

f(a,b, c) = a+b+c
g(a,b,c) = (a+bw+cw^2)
and h(a,b,c) = (a+bw^2+cw)


then a^3+ b^3+ c^3-3abc = f(a,b,c) g(a,b,c) h(a,b,c)

and x^3+ y^3+ z^3-3xyz = f(x,y,z) g(x,y,z) h(x,y,z)

so (a^3+ b^3+ c^3-3abc)( x^3+ y^3+ z^3-3xyz) = f(a,b,c) g(a,b,c) h(a,b,c) f(x,y,z) g(x,y,z) h(x,y,z)

now let us evaluate f(a,b,c) f(x,y,z) , g(a,b,c) g(x,y,z) and h(a,b,c) h(x,y,z)

f(a,b,c) f(x,y,z) = (a+b+c)(x+y+z) = (ax+ay+ az + bx +by + bz + cx + cy+ cz)

g(a,b,c) g(x,y,z) = (a+bw+cw^2) (x+yw+zw^2)

= (ax +ayw+azw^2 + bxw + byw^2 + bzw^3 + cxw^2 + cyw^3 + czw^4)
= (ax +ayw+azw^2 + bxw + byw^2 + bz + cxw^2 + cyw^3 + czw) knowing w^3 = 1 and hence w^4 = w
= (ax +bz+cy + (ay + bx + cz)w+ (az + by+cx) w^2)
similarly
h(a,b,c) h(x,y,z) = (a+cw+ bw^2) (x+zw+ yw^2)

= (ax+bz+cy) + (az+by+cx)w+ (ay+bx+cz) w^2

If we put ax + bz+ cy = p, ay+bx+cz = q, az+by+cx = r

We get

f(a,b,c) f(x,y,z) = p + q + r = f(p,q,r)

g(a,b,c) g(x,y,z) = p + qw + r w^2 = g(p,q,r)

h(a,b,c) h(x,y,z) = p + rw + q w^2 = h(p,q,r)

so (a^3+ b^3+ c^3-3abc)( x^3+ y^3+ z^3-3xyz) = f(a,b,c) g(a,b,c) h(a,b,c) f(x,y,z) g(x,y,z) h(x,y,z) = f(p,q,r) g(p,q,r) h(p,q,r) = ( p^3+ q^3+ r^3-3pqr)

hence proved