Wikipedia:Reference desk/Archives/Mathematics/2016 August 8

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August 8

b^a=ab ?

$5^{2}=25$ and $1^{0}=01$. Are there other examples where $b^{a}$ is written $ab$ in decimal notation? Bo Jacoby (talk) 14:53, 8 August 2016 (UTC).

• My brute-force script says no for $a\leq 20,b\leq 10000$. That is not an elegant solution, but it seems easy to prove that there is no such pair of integers for a>1 and b sufficiently large (the number of digits alone will not match), and the a=0,1 cases are trivial to do. Tigraan^{Click here to contact me} 15:14, 8 August 2016 (UTC)

If you allow different bases then you get a number of other examples, such as 3³ = 33 (base 8), 4³=34 (base 20). You get a solution for a=2 exactly when the base is a triangular number (such as 10). From sketching the graph it appears that there are only a finite number of solutions for a given base, assuming a and b are non negative integers. If you allow negative integers then b^a=10a+b also has the solution b=-4, a=2. --RDBury (talk) 19:20, 8 August 2016 (UTC)