Q) Show that there cannot be distinct positive integers \(a,b,c,d\) such that \(a+b=c+d\) and \(a^3+b^3=c^3+d^3\).
Solution:
This problem trivially arrives to a scenario that we have covered in the module ‘System of Equations’. And it touches a very important concept that is useful in many scenarios.
We have \(a+b=c+d\). Cubing both sides, and cancelling out the common terms (assuming that \(a^3+b^3=c^3+d^3\) holds), we arrive at \(ab=cd\). The important thing is to realize that 4 distinct integers can’t have the same sum and the same product.
The reason lies in the theory of roots of a quadratic equation. Let’s have \(a+b=c+d=S\) and \(ab=cd=P\). Now consider the equation \(X^2-SX+P=0\). We have the roots of this equation as \((a,b)\) and \((c,d)\). This is impossible, thus invalidating our assumption that \(a^3+b^3=c^3+d^3\) shall hold with \(a+b=c+d\).
Mind you, that this flow would have stayed the same if the sum of the squares would have been equated instead of the sum of the cubes. An open question, in this context (and I urge all of you to try that and think about that) would be to analyse if this holds true for higher powers as well.
In other words – can we have \(a+b=c+d\) and \(a^k+b^k=c^k+d^k\) for some \(k \in \mathbb{N}\)?