Concept 1 Perfect Squares

Concept I Perfect Squares

There has been a huge surge in the number of questions about perfect squares, in almost all mocks. The basic trick to any such question is assuming the number as a perfect square of an integer k and then using techniques of completion of square and then the formula of (a^2-b^2) and solving using divisibility theory

Example I Find all natural n such that n(n+16) is a perfect square

step 1 n(n+16)=k^2

step 2 (n^2+2.8.n+8^2)-8^2=k^2

step3 (n+8+k)(n+8-k)=64

see now lhs and rhs both are integers then both of (n+8-k) and (n+8+k) are divisors of 64. But note that we add the two equations we will get 2n+16, so teh sum of two divisors should be even hence both divisors even or both odd

so n+8+k=32,16,8,4,2 and n+8-k=2,4,8,16,32

but see this n is positive hence k is positive, thus n+8+k>n+8-k
so only two options

and solving we get 2n+16=34,20
so n=9,2

Note : The source of this problem is Pomona Wisconsin mathematics talent search exam!

Practice problem!!
Find the sum of all such positive integers m's such that m^2+25m+19 is a perfect square

Now we will extend the method to other kinds of problems
Basically what we used in the above problem is difference of square method

lets take an example
x^6=y^2+127, find the no of pairs of postive integers (x,y)

first step in this problem is recognising that 127 is a prime
then we move to
(x^3+y)(x^3-y)=127
so clearly 2x^3=128 x=4 and y=63

so one pair (4,63)