Question

Find all integers a,b (a is distinct from b) such that a^2 - b^2 =(a-b)^5 , and a^2 , b^2 < 100

Answer 1

help me !

Answer 2

a^2 -b ^2 = (a+b)(a-b)

Answer 3

I don't know how much it'll simplify after that.

Note that we can cancel a-b from both sides since a is not equal to b

Answer 4

a+b = (a-b)^4

We are given -10<a<10 and -10<b<10

Here we can see a+b has to be 4th power of some integer
Also, max value of a+b = 20
Also, a+b has to be positive , thus a>b

Using all this, we can say a+b will be one of 1 or 16, and a-b one of 1 or 2

Answer 5

Wait, my last hypothesis is false, a is not necessarily greater than b
so a+b can be 1 or 16
a-b can be -1,1,-2 or 2

Answer 6

for a+b =1
a-b = 1 or -1

1st case : a=1 and b=0
2nd case : a=0 and b=-1

for a+b =16
a-b = 2 or -2

1st case, a=9 and b=7
2nd case : a= 7 and b=9

Answer 7

Sorry in 2nd case when a+b =1 and a-b =-1,b should be 1 and not -1

Answer 8

So final possible pairs are
(1,0)(0,1)(7,9)(9,7)

Answer 9

ohh thaks shub