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    <p>You can make it run slightly faster by avoiding the while loops in favor of setting low to 10 ** (len(str(x)) / n) and high to low * 10. Probably better is to replace the len(str(x)) with the bitwise length and using a bit shift. Based on my tests, I estimate a 5% speedup from the first and a 25% speedup from the second. If the ints are big enough, this might matter (and the speedups may vary). Don't trust my code without testing it carefully. I did some basic testing but may have missed an edge case. Also, these speedups vary with the number chosen.</p> <p>If the actual data you're using is much bigger than what you posted here, this change may be worthwhile.</p> <pre><code>from timeit import Timer def find_invpow(x,n): """Finds the integer component of the n'th root of x, an integer such that y ** n &lt;= x &lt; (y + 1) ** n. """ high = 1 while high ** n &lt; x: high *= 2 low = high/2 while low &lt; high: mid = (low + high) // 2 if low &lt; mid and mid**n &lt; x: low = mid elif high &gt; mid and mid**n &gt; x: high = mid else: return mid return mid + 1 def find_invpowAlt(x,n): """Finds the integer component of the n'th root of x, an integer such that y ** n &lt;= x &lt; (y + 1) ** n. """ low = 10 ** (len(str(x)) / n) high = low * 10 while low &lt; high: mid = (low + high) // 2 if low &lt; mid and mid**n &lt; x: low = mid elif high &gt; mid and mid**n &gt; x: high = mid else: return mid return mid + 1 x = 237734537465873465 n = 5 tests = 10000 print "Norm", Timer('find_invpow(x,n)', 'from __main__ import find_invpow, x,n').timeit(number=tests) print "Alt", Timer('find_invpowAlt(x,n)', 'from __main__ import find_invpowAlt, x,n').timeit(number=tests) </code></pre> <p>Norm 0.626754999161</p> <p>Alt 0.566340923309</p>
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    1. POHow to compute the nth root of a very big integer
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    1. COYour second function, find_invpowAlt, gives a wildly wrong answer for x=118997879821732370764604711647724283139870175351576755860556891902958645241483485254092600557474860904935286687480039428945219115513349647465379580432922136155355040992635166676363150438436216219094913514982415747153956476970303302126880391024128871557664284712411567099374094385902892603751471822837746770111, n=3
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