[Дзен-трансгуманист]

Deff,
Сравним скорость?

// ------------------------------ Моё ------------------------------

var factorize = ( function ( maxNumber ) {

  var primesMapSize = Math.ceil( Math.sqrt( maxNumber ) );
  var primesMap = [ void 0, true ];
  var primesList = [];

  for ( var number = 2; number <= primesMapSize; number++ ) {

    if ( primesMap[ number ] === false ) {
      continue;
    }

    primesList.push( number );
    primesMap[ number ] = true;

    for ( var next = number * 2; next <= primesMapSize; next += number ) {
      primesMap[ next ] = false;
    }
  }

  maxNumber = primesMapSize * primesMapSize;

  return function ( number ) {

    if ( !isFinite( number = +number ) || number < 1 || number != Math.floor( number ) ) {
      return Number.NaN;
    }
    if ( number > maxNumber ) {
      return false;
    }
    if ( number == 1 ) {
      return [];
    }
    if ( number <= primesMapSize && primesMap[ number ] == true ) {
      return [ { factor : number, times : 1 } ];
    }

    var maxPrime = Math.floor( Math.sqrt( number ) );
    var prime;
    var primeId = 0;
    var factor;
    var factors = [];

    while ( ( prime = primesList[ primeId++ ] ) <= maxPrime ) {

      if ( number % prime == 0 ) {

        factors.push( factor = { factor : prime, times : 1 } );

        while ( ( number /= prime ) % prime == 0 ) {
          factor.times++;
        }

        if ( number == 1 ) {
          return factors;
        }

        maxPrime = Math.floor( Math.sqrt( number ) );
      }
    }

    factors.push( { factor : number, times : 1 } );
    return factors;
  };

})( Math.pow( 2, 40 ) );

function normalizeSqrt ( number ) {

  var factors = factorize( number );

  if ( typeof factors == "number" && isNaN( factors ) ) {
    return "invalid number";
  }
  if ( factors === false ) {
    return "number too big";
  }

  var rational = 1;
  var irrational = 1;
  var factor;

  while ( factor = factors.pop() ) {

    rational *= Math.pow( factor.factor, factor.times >> 1 );

    if ( factor.times & 1 ) {
      irrational *= factor.factor;
    }
  }

  return rational + " * sqrt( " + irrational + " )";
}

// ------------------------------ Deff'а ------------------------------

function Tstsqrt(Z) {
   var a = Math.floor(Math.sqrt(Z));
   if(a<2) return false;
   for(var i=a; i>=2; i--){
      var j=Z/(i*i);
      if(Math.floor(j)==j){
         return (i+' * sqrt( '+j+' )');
      }
   } return false;
}

// ------------------------------ Бенчмарк ------------------------------

var numbers = [
  906972642240,
  1099505336329,
  1099505336330,
  1099505336331,
  1099505336332,
  1099505336333,
  Math.pow( 2, 40 ) - 1
];

function test ( title, func ) {

  var result = title + "\n\n";
  var timeStart = +new Date();

  for ( var i = 0; i < numbers.length; i++ ) {
    result += numbers[ i ] + " : " + func( numbers[ i ] ) + "\n";
  }

  result += "\nTime: " + ( ( +new Date() - timeStart ) / 1000 ).toFixed( 3 ) + " sec.";
  return result;
}

alert( test( "normalizeSqrt", normalizeSqrt ) + "\n\n\n\n" + test( "Tstsqrt", Tstsqrt ) );