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Is there any known integer exponentiation algorithm to compute $x^y$ for the special case $x = 3$ which is faster than the general case algorithm found in [1], section 4.6.3?

[1] D. E. Knuth, The Art of Computer Programming. Volume 2: Seminumerical Algorithms, Addison-Wesley, 1981

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Depends. How do you want the result? If trinary notation is acceptable, you can get it in linear time by outputting an 1 followed by $y$ 0s. –  Henning Makholm Nov 27 '11 at 15:53
    
Is Knuth talking about the squaring algorithm in Section 4.6.3 of the Volume 2? –  Simone Nov 27 '11 at 17:01
    
@Henning: That's just silly! Because it works for any $x$, doesn't it? Not just $x=3$. "Give me $x^y$ in base $x$". –  TonyK Nov 27 '11 at 17:27
    
@Simone: I believe so even though I don't have the referenced print physically available. –  jnml Nov 27 '11 at 17:54
    
@TonyK: Yes, it's a bit silly -- but there's a point to the silliness, namely that the achievable efficiency depends on which representation is required for the result, so the OP should not expect a precise answer without specifying a representation first. –  Henning Makholm Nov 27 '11 at 21:20

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