Let a be the square root of 2 and b the square root of 3. Prove that there is no set of integers m,n,p except 0,0,0 for which m+ns+pb=0.

solution:

If m+na+pb=0, then

na+pb=-m,

(na+pb)(na+pb)=m²,

2n²+3p²+2npc=m²,

where c is the square root of 6.

Therefore

c=(m²-2n²-3p²)/2np.

Since c is irrational, we have np=0, and therefore n=0 or p=0.

If p=0, then m+na+pb=0 yields na=-m and since a is irrational we have m=0 and n=0.

Similar reasoning applies to the case n=0.