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

= 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 ba=10a+b also has the solution b=-4, a=2. --RDBury (talk) 19:20, 8 August 2016 (UTC)