Quant Boosters  Soumya Chakraborty  Set 3

a^2 + ab + b^2 = n^2
(a+b)^2 = n^2 + ab
(a+b+n)(a+bn) = abas 'a' and 'b' are prime numbers
ab has 4 factors: 1, a, b, ab
obviously, a + b + n > a + b  nCase 1:
a + b + n = ab
a + b  n = 1Case 2:
a + b + n = a
a + b  n = b ... considering a>bBut, case 2 is not possible.
b = n
a = n,
a = b ... which is impossibleCase 1:
adding the two equations
2a + 2b = ab + 1
2a  ab + 2b = 1
a(2b)  2(2b) = 3
(a2)(b2) = 3only possibility for 'a' and 'b' being prime and satisfying the above condition is: 5 and 3
giving us he only possibility 49