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Grade: 10

                        

find all pairs of positive integers a,b such that a^b-b^a=3

3 years ago

Answers : (1)

Arun
24742 Points
							
 
Since RHS is odd, we must have either a is even and b is odd or vice versa.
 
Case I: a is even, b is odd
 
Also if b \ne 1, we must haveclearly, b>a. Hence a \ge 2 and hence b \ge 3. (In fact looking modulo 3, a is an even number of the form 3k+1, and hence the least value of a is 4, but that is not required here)
 
Now a^b will be of the form 8k, whereas b^a being the square of an odd number will be of the form 8k+1. So in the equationa^b = b^a+3, LHS is of the form 8k while RHS is of the form 8k+4, which is a contradiction.
 
Case II: a is odd, b is even. Again, a \ge 3, So in the equation a^b = b^a+ 3, LHS is of the form 8k+1 while RHS is of the form 8k+3 and so no solutions are possible.
 
That only leaves us with the possibility, b =1, in which case a = 4. Hence (4,1) is the only solution in natural numbers.
3 years ago
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