MathLabs

Problem 2

Show that (36a+b)(a+36b)(36a+b)(a+36b) cannot be a power of 22 for any positive integers aa and bb.
Step 3 of 4: Descend by halving
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a=2a′,b=2b′a=2a',\qquad b=2b'
Detailed analysis

In particular a,ba,b are even. Put a=2a′a=2a' and b=2b′b=2b'. Then (36a+b)(a+36b)=4(36a′+b′)(a′+36b′)(36a+b)(a+36b)=4(36a'+b')(a'+36b'), so the smaller pair (a′,b′)(a',b') is another counterexample.