Mathematics Proof

Question

Prove that 67n+17 cannot be a perfect square for any three consecutive integers n.

Answer 1

67n+17 cannot be a perfect square for 3 consecutive integers

square root of (67n+17)

not a perfect square for n=0, 1, 2, 3

assume it's not for n=k

try to show it's not for n=k+1

proof by mathematical induction

67k+17 is not a perfect square for any integer k

try to show 67(k+1)+17 is not a perfect square

67k+67+17

= 67k+ 17 +67

67 is not a perfect square

67k+17 is not a perfect square

their sum is not a perfect square

a^2+b^2=c^2 = hypotenuse of a right triangle

sum of 2 irrationals cannot= an integer