Question

Let a and b be natural numbers such that 2a – b, a – 2b, and a + b are all distinct squares. What is the smallest possible value of b?

Answer 1

Let

2a – b = n² …(i)

a – 2b = p² …(ii)

a + b = k² …(iii)

Adding (ii) and (iii) we get

2a – b = p² – k²

p² + k² = n² (p < k as a + b < a – 2b)

For b to be smallest, k² and p² is also small and

must be multiple of 3 (as 3b = k² – p²)

For smallest b, the least value of k and p be 12 and 9 resp.

∴ Least value of b = 21