Integer roots

Integer roots is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Integer Roots

The task is to write a program that computes the Nth root of X as the closest integer less than or equal to R for which R^N=X. N and X are both integers.

As a test, you can calculate the first 2,001 digits in the decimal expansion of the square root of two. The method involves multiplying 2 by 100^2000 before taking the square root. You will then have the 2,001 most significant digits of the square root of two. Just remember where the decimal point goes.

Python

<lang python>def root(a,b):

   a-=1
   c=1
   if c==0:c=1
   d=(a*c+b//(c**a))//(a+1)
   if d==0:d=1
   e=(a*d+b//(d**a))//(a+1)
   if e==0:e=1
   while c!=d and c!=e:
       c,d,e=d,e,(a*e+b//(e**a))//(a+1)
   return min(d,e)

print(root(2,2*100**2000))</lang>

Output:

141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008