Lesson How many integers of the form n^2 + 18n + 13 are perfect squares

How many integers of the form n^2 + 18n + 13 are perfect squares

Problem 1

Let  n^2 + 18n + 13 = m^2   be a perfect square.


Then


    (n + 9)^2 - 68 = m^2

    (n + 9)^2 - m^2 = 68

    (n + m + 9)*(n - m + 9) = 68.


Decompositions for 68 are  1*68, 2*34, 4*17, 17*4, 34*2 and 68*1.


For each decomposition, we have the system of equations


    n + m + 9 =  1
    n - m + 9 = 68


    n + m + 9 =  2
    n - m + 9 = 34


    n + m + 9 =  4
    n - m + 9 = 17


    n + m + 9 = 17
    n - m + 9 =  4


    n + m + 9 = 34
    n - m + 9 =  2


    n + m + 9 = 68
    n - m + 9 =  1


Easy analysis shows that some of these systems produce non-integer solutions, that are out of our scope.


The only system, which produces appropriate integer solutions, is THIS


    n + m + 9 = 34
    n - m + 9 =  2


The solution is  n = 9, m = 16.


Therefore,  n = 9 is the solution to the problem.


If you analyze the same systems with decomposition of the number 68 into the product of negative factors, 
you will find another solution n = -27.


In all, there are 2 (two) such numbers n, 9 and -27.

Problem 2

Consider any integer positive number  " n ",  which has the last digit of 2.


(In other words, consider any integer positive number n such that  n = 2 (mod 10) ).


For example, such numbers "n" are 12, 22, 32, 42, 52 . . . and so on.


Then the number  19n  has the last digit of  8,  and the number  (19n+9)  has the last digit of 7,  because  8 + 9 = 17 = 7 (mod 10).


    +-------------------------------------------------------------------------+
    |   But  NO  ONE  such integer positive number is a perfect square  (!)   |
    +-------------------------------------------------------------------------+



(Notice that a square of an integer number may have the last digit only 0, 1, 4, 5, 6, 9, but may not have the last digit of 2, 3, 7, 8).


Therefore, all the numbers  (19n +9)  with n = 2 (mod 10) are not perfect squares and, THEREFORE, create/produce  IRRATIONAL numbers .