MathLabs

Problem 5

Find all pairs (a,b)(a,b) of positive integers satisfying ab2=ba a^{b^2}=b^a .
Step 3 of 5: Exclude d≥0 d\ge0
In plain words

Exclude d≥0 d\ge0

d=2n−md=2n-m
Detailed analysis

Rearrange as n/m=sd n/m=s^d . If d=0 d=0, then m=n m=n and the equation gives m=2m m=2m , impossible. If d>0 d>0, then n>m n>m , so d=2n−m>n d=2n-m>n ; but n=msd≥sd>d n=ms^d\ge s^d>d , impossible.