MathDB

An integer

c

is square-friendly if it has the following property: For every integer

m

, the number

m^2+18m+c

is a perfect square. (A perfect square is a number of the form

n^2

, where

n

is an integer. For example,

49 = 7^2

is a perfect square while

46

is not a perfect square. Further, as an example,

6

is not square-friendly because for

m = 2

, we have

(2)2 +(18)(2)+6 = 46

, and

46

is not a perfect square.) In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following: (a) Find a square-friendly integer, and prove that it is square-friendly. (b) Prove that there cannot be two different square-friendly integers.

Problem Statement