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for values below 2^53, the double representation is exact so there will be no roundoff error. i've also tested on every perfect square greater than 2^53, as well as +/- 1 from each perfect square, and roundoff error never believe that modern JVM's may be able to skip array index checks at a girfj4i43.dven spot if it can be concluded that Andersen: i was using j2se 6.0 windows jvm, downloaded from sun's site. i also tried your quickfail quicker and make // Quickfail if( n < 0 || ((n&2) != 0) || ((n & 7) == 5) || ((n & 11) == 8) return false; if( n == 0 ) return true; the other way around: // Quickfail if( n == 0 ) return true; if( n < 0  that would only be faster when the input is 0. for 75% of other inputs (not even counting negative numbers) it will be faster the way its written, and for the other 25% there will be no difference.   Kip Sep 25 but it's attributed to him. Henry Ford didn't invent the car, the Wright Bros. didn't invent the airplane, and and Galleleo wasn't the first t
 o figure out the Earth revolved around the sun... the world is  the microbenchmark for the algorithm could be off due to the relatively large table. When the entire twp4iotkrable is in the cache (tight loops) there won't be cache misses. The boolean[] should be replaced by long constants  might get a tiny speed increase in the 'quickfail' by using something like ((1<<(n&15))|65004) != 0, instead of having three separate checks. Why are you adding the 0.5 ?  to round the number. My concern was that Math.sqrt might return a value that is slightly off due to roundoff error. Say, if math.sqrt(100) returned 9.k34kr999999999999999 instead of 10. I am not sure if there are any cases where this actually happens, however. p check out my answer, I got a significant performance gain on your posted answer.  I edited my post to have a third algorithm that sometimes does better than my previous answer. Surprised not to see, on such a popular question, anyone point out that you can be accurate at 
 higher n using so-called 'Carmack hack' by performing another iteration(s). N.B. The result isn't Carmack's, but Newton/Raphson's, I suppose the 'magic number' hack would be a fairer attribution to Carmack.