Question

Let m, n be positive integers. If m³ + n³ is the square of an integer, then prove that (m + n) is not a product of two different prime numbers.

Answer 1

Let us assume that m + n = pq, where p and q are distinct prime.

Since m³ + n³ = (m + n) (m² – mn + n²)

= (m + n) [(m + n)² – 3mn]

is a square so m³ + n³ = (m + n) [(m + n)² – 3mn]

must divisible by pq.

=> 3 mn must divisible by p and q.

Since p ≠ q let q ≠ 3 so q | m but q | m+ n so q| n

m = qx, n = qy

=> p = 3.

so m + n = pq = 3q

=> qx + qy = 3q

=> m = 2q, n = q

or m = q, n = 2q

m³ + n³ = 9q³ is not a square of an integer which is a contradiction

Hence m + n is not a product of two different prime.