MathLabs

Problem 5

Find all pairs (a,b)(a,b) of positive integers satisfying ab2=ba a^{b^2}=b^a .
Step 2 of 5: Use a common primitive base
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In plain words

Use a common primitive base

a=sm,b=sna=s^m,\quad b=s^n
Detailed analysis

Write each integer uniquely as a power of a non-perfect-power integer s>1 s>1: a=sm a=s^m and b=sn b=s^n . Equality of the prime factorizations in ab2=ba a^{b^2}=b^a forces the bases to agree. Comparing exponents gives ms2n=nsm m s^{2n}=n s^m .