Show that the equation has no integral root for any integer n.
We note that implies is divisible by 7. This implies x is divisible by 7 (as 7 is a prime number). Suppose x= 7x’. Hence we can rewrite the given equation as:
Cancelling out a 7 we have . Since 7 divides left hand side, it must also divide the right hand side. Since 7 cannot divide 2, it must divide as 7 and 2 are coprime. Note that 7 cannot divide as square of a number always gives remainder 0, 1, 4, 2 when divided by 7 and never 6. But if is divisible by 7 then must give remainder 6 when divided by 7. Hence contradiction.
Necessary Lemma: square of a number always gives remainder 0, 1, 4, 2 when divided by 7
Key Ideas: Modular Arithmetic