I did some searching and was not able to find any information regarding this implementation versus every other one I have seen.
function sieve($top)
{
for($i = 11; $i<$top; $i+=2)
{
if($i % 3 == 0 || $i % 5 == 0
|| $i % 7 == 0)
{
continue;
}
echo "$i <br />";
}
}
Yeah I know it just prints it out, but that's not the important part. What is the major pitfall whether it be time or other?
EDIT: Are there any other issues beyond scalability? Also again thanks for the comments about moving forward with prime finding.
The major pitfall of this is it doesn't scale. Once the numbers are large enough anything will be returned. You list of modulus excluders needs to grow with the search.
You can refer to Sieve of Eratosthenes on Wikipedia; and this link for a PHP implementation.
It's limited to prime numbers up to 11. To extend it any further you need to add || $u % 11 == 0 || $i % 13 == 0 ... etc
First, you're only checking against three numbers (3, 7, and 11). For the Sieve of Erathosthenes, you should start with a list of numbers, 2..i. Then loop through that list, and remove numbers that are factors of the number you're iterating on. For example, once you get to 7, which is prime, you'll need to remove 49, 56, and other multiples of 7.
Second, the method I just described would scale very poorly - if you tried looking for primes from 1..10^9, you'd need 10^9 values in your list. There are other ways besides the Sieve of Erathosthenes to find prime numbers - see http://en.wikipedia.org/wiki/Prime_number
This function use "Sieve of Eratosthenes Algorithm"
function getPrimaryNumbers($n)
{
$sieve = [];
for($i = 1; $i <= $n; $i++) {
$sieve[$i] = $i;
}
$i =2;
while($i * $i <= $n) {
if(isset($sieve[$i])) {
$k = $i;
while ($k * $i <= $n) {
unset($sieve[$k * $i]);
$k++;
}
}
$i++;
}
return $sieve;
}
Related
If you have a simple counter loop, how do you detect special patterns, for example, every 10 increments but at 6/16/26/36. $i needs to start at 0 too.
The only approach I can think of is this one, but obviously it doesn't work for large loops:
for ($i=0; $i < 1000; $i++) {
// if ( $i == 6 || $i == 16 || $i == 26...... etc ) { do something }
}
There's not going to be one single answer for all types of patterns, but so long as there is a pattern, you can figure it out:
for ($i=0; $i<1000; $i++) {
if (($i-6)%10 == 0) {
// every time $i minus 6 is evenly divisible by 10
}
}
I'm struggling with Project Euler problem 23: Non-abundant sums.
I have a script, that calculates abundant numbers:
function getSummOfDivisors( $number )
{
$divisors = array ();
for( $i = 1; $i < $number; $i ++ ) {
if ( $number % $i == 0 ) {
$divisors[] = $i;
}
}
return array_sum( $divisors );
}
$limit = 28123;
//$limit = 1000;
$matches = array();
$k = 0;
while( $k <= ( $limit/2 ) ) {
if ( $k < getSummOfDivisors( $k ) ) {
$matches[] = $k;
}
$k++;
}
echo '<pre>'; print_r( $matches );
I checked those numbers with the available on the internet already, and they are correct. I can multiply those by 2 and get the number that is the sum of two abundant numbers.
But since I need to find all numbers that cannot be written like that, I just reverse the if statement like this:
if ( $k >= getSummOfDivisors( $k ) )
This should now store all, that cannot be created as the sum of to abundant numbers, but something is not quit right here. When I sum them up I get a number that is not even close to the right answer.
I don't want to see an answer, but I need some guidelines / tips on what am I doing wrong ( or what am I missing or miss-understanding ).
EDIT: I also tried in the reverse order, meaning, starting from top, dividing by 2 and checking if those are abundant. Still comes out wrong.
An error in your logic lies in the line:
"I can multiply those by 2 and get the number that is the sum of two abundant numbers"
You first determine all the abundant numbers [n1, n2, n3....] below the analytically proven limit. It is then true to state that all integers [2*n1, 2*n2,....] are the sum of two abundant numbers but n1+n2, and n2+n3 are also the sum of two abundant numbers. Therein lies your error. You have to calculate all possible integers that are the sum of any two numbers from [n1, n2, n3....] and then take the inverse to find the integers that are not.
I checked those numbers with the available on the internet already, and they are correct. I can multiply those by 2 and get the number that is the sum of two abundant numbers.
No, that's not right. There is only one abundant number <= 16, but the numbers <= 32 that can be written as the sum of abundant numbers are 24 (= 12 + 12), 30 (= 12 + 18), 32 (= 12 + 20).
If you have k numbers, there are k*(k+1)/2 ways to choose two (not necessarily different) of them. Often, a lot of these pairs will have the same sum, so in general there are much fewer than k*(k+1)/2 numbers that can be written as the sum of two of the given k numbers, but usually, there are more than 2*k.
Also, there are many numbers <= 28123 that can be written as the sum of abundant numbers only with one of the two abundant numbers larger than 28123/2.
This should now store all, that cannot be created as the sum of to abundant numbers,
No, that would store the non-abundant numbers, those may or may not be the sum of abundant numbers, e.g. 32 is a deficient number (sum of all divisors except 32 is 31), but can be written as the sum of two abundant numbers (see above).
You need to find the abundant numbers, but not only to half the given limit, and you need to check which numbers can be written as the sum of two abundant numbers. You can do that by taking all pairs of two abundant numbers (<= $limit) and mark the sum, or by checking $number - $abundant until you either find a pair of abundant numbers or determine that none sums to $number.
There are a few number theoretic properties that can speed it up greatly.
Below is php code takes 320 seconds
<?php
set_time_limit(0);
ini_set('memory_limit', '2G');
$time_start = microtime(true);
$abundantNumbers = array();
$sumOfTwoAbundantNumbers = array();
$totalNumbers = array();
$limit = 28123;
for ($i = 12; $i <= $limit; $i++) {
if ($i >= 24) {
$totalNumbers[] = $i;
}
if (isAbundant($i)) {
$abundantNumbers[] = $i;
}
}
$countOfAbundantNumbers = count($abundantNumbers);
for ($j = 0; $j < $countOfAbundantNumbers; $j++) {
if (($j * 2) > $limit)
break; //if sum of two abundant exceeds limit ignore that
for ($k = $j; $k < $countOfAbundantNumbers; $k++) { //set $k = $j to avoid duble addtion like 1+2, 2+1
$l = $abundantNumbers[$j] + $abundantNumbers[$k];
$sumOfTwoAbundantNumbers[] = $l;
}
}
$numbers = array_diff($totalNumbers, $sumOfTwoAbundantNumbers);
echo '<pre>';print_r(array_sum($numbers));
$time_end = microtime(true);
$execution_time = ($time_end - $time_start);
//execution time of the script
echo '<br /><b>Total Execution Time:</b> ' . $execution_time . 'seconds';
exit;
function isAbundant($n) {
if ($n % 12 == 0 || $n % 945 == 0) { //first even and odd abundant number. a multiple of abundant number is also abundant
return true;
}
$k = round(sqrt($n));
$sum = 1;
if ($n >= 1 && $n <= 28123) {
for ($i = 2; $i <= $k; $i++) {
if ($n % $i == 0)
$sum+= $i + ( $n / $i);
if ($n / $i == $i) {
$sum = $sum - $i;
}
}
}
return $sum > $n;
}
function find_highest_prime_factor($n)
{
for ($i = 2; $i <= $n; $i++)
{
if (bcmod($n, $i) == 0) //its a factor
{
return max($i, find_highest_prime_factor(bcdiv($n,$i)));
}
}
if ($i == $n)
{
return $n; //it's prime if it made it through that loop
}
}
UPDATE: This is the correct answer, my bad!
Get rid of the final if statement otherwise if $i!=sqrt($n) because sqrt($n) is not an integer you have an undefined return value
function find_highest_prime_factor($n){
for ($i = 2; $i <= sqrt($n); $i++) //sqrt(n) is the upperbound
{
if (bcmod($n, $i) == 0) //its a factor
{
return max($i, find_highest_prime_factor(bcdiv($n,$i)));
}
}
return $n; //it's prime if it made it through that loop
}
Line 11 should be:
if ($i == ceil(sqrt($n)))
Starting at 2 and stepping by 1 is inefficient. At least check 2 separately and then loop from 3 stepping 2 each time. Using a 2-4 wheel would be even faster.
When you recurse your code starts again at trial factor 2. It would be better to pass a second parameter holding the factor you have reached so far. Then the recursion wouldn't have go back over old factors that have already been tested.
I'm attempting to solve Project Euler in PHP and running into a problem with my for loop conditions inside the while loop. Could someone point me towards the right direction? Am I on the right track here?
The problem, btw, is to find the sums of all prime numbers below 2,000,000
Other note: The problem I'm encountering is that it seems to be a memory hog and besides implementing the sieve, I'm not sure how else to approach this. So, I'm wondering if I did something wrong in the implementation.
<?php
// The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
// Additional information:
// Sum below 100: 1060
// 1000: 76127
// (for testing)
// Find the sum of all the primes below 2,000,000.
// First, let's set n = 2 mill or the number we wish to find
// the primes under.
$n = 2000000;
// Then, let's set p = 2, the first prime number.
$p = 2;
// Now, let's create a list of all numbers from p to n.
$list = range($p, $n);
// Now the loop for Sieve of Eratosthenes.
// Also, let $i = 0 for a counter.
$i = 0;
while($p*$p < $n)
{
// Strike off all multiples of p less than or equal to n
for($k=0; $k < $n; $k++)
{
if($list[$k] % $p == 0)
{
unset($list[$k]);
}
}
// Re-initialize array
sort ($list);
// Find first number on list after p. Let that equal p.
$i = $i + 1;
$p = $list[$i];
}
echo array_sum($list);
?>
You can make a major optimization to your middle loop.
for($k=0; $k < $n; $k++)
{
if($list[$k] % $p == 0)
{
unset($list[$k]);
}
}
By beginning with 2*p and incrementing by $p instead of by 1. This eliminates the need for divisibility check as well as reducing the total iterations.
for($k=2*$p; $k < $n; $k += $p)
{
if (isset($list[k])) unset($list[$k]); //thanks matchu!
}
The suggestion above to check only odds to begin with (other than 2) is a good idea as well, although since the inner loop never gets off the ground for those cases I don't think its that critical. I also can't help but thinking the unsets are inefficient, tho I'm not 100% sure about that.
Here's my solution, using a 'boolean' array for the primes rather than actually removing the elements. I like using map,filters,reduce and stuff, but i figured id stick close to what you've done and this might be more efficient (although longer) anyway.
$top = 20000000;
$plist = array_fill(2,$top,1);
for ($a = 2 ; $a <= sqrt($top)+1; $a++)
{
if ($plist[$a] == 1)
for ($b = ($a+$a) ; $b <= $top; $b+=$a)
{
$plist[$b] = 0;
}
}
$sum = 0;
foreach ($plist as $k=>$v)
{
$sum += $k*$v;
}
echo $sum;
When I did this for project euler i used python, as I did for most. but someone who used PHP along the same lines as the one I did claimed it ran it 7 seconds (page 2's SekaiAi, for those who can look). I don't really care for his form (putting the body of a for loop into its increment clause!), or the use of globals and the function he has, but the main points are all there. My convenient means of testing PHP runs thru a server on a VMWareFusion local machine so its well slower, can't really comment from experience.
I've got the code to the point where it runs, and passes on small examples (17, for instance). However, it's been 8 or so minutes, and it's still running on my machine. I suspect that this algorithm, though simple, may not be the most effective, since it has to run through a lot of numbers a lot of times. (2 million tests on your first run, 1 million on your next, and they start removing less and less at a time as you go.) It also uses a lot of memory since you're, ya know, storing a list of millions of integers.
Regardless, here's my final copy of your code, with a list of the changes I made and why. I'm not sure that it works for 2,000,000 yet, but we'll see.
EDIT: It hit the right answer! Yay!
Set memory_limit to -1 to allow PHP to take as much memory as it wants for this very special case (very, very bad idea in production scripts!)
In PHP, use % instead of mod
The inner and outer loops can't use the same variable; PHP considers them to have the same scope. Use, maybe, $j for the inner loop.
To avoid having the prime strike itself off in the inner loop, start $j at $i + 1
On the unset, you used $arr instead of $list ;)
You missed a $ on the unset, so PHP interprets $list[j] as $list['j']. Just a typo.
I think that's all I did. I ran it with some progress output, and the highest prime it's reached by now is 599, so I'll let you know how it goes :)
My strategy in Ruby on this problem was just to check if every number under n was prime, looping through 2 and floor(sqrt(n)). It's also probably not an optimal solution, and takes a while to execute, but only about a minute or two. That could be the algorithm, or that could just be Ruby being better at this sort of job than PHP :/
Final code:
<?php
ini_set('memory_limit', -1);
// The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
// Additional information:
// Sum below 100: 1060
// 1000: 76127
// (for testing)
// Find the sum of all the primes below 2,000,000.
// First, let's set n = 2 mill or the number we wish to find
// the primes under.
$n = 2000000;
// Then, let's set p = 2, the first prime number.
$p = 2;
// Now, let's create a list of all numbers from p to n.
$list = range($p, $n);
// Now the loop for Sieve of Eratosthenes.
// Also, let $i = 0 for a counter.
$i = 0;
while($p*$p < $n)
{
// Strike off all multiples of p less than or equal to n
for($j=$i+1; $j < $n; $j++)
{
if($list[$j] % $p == 0)
{
unset($list[$j]);
}
}
// Re-initialize array
sort ($list);
// Find first number on list after p. Let that equal p.
$i = $i + 1;
$p = $list[$i];
echo "$i: $p\n";
}
echo array_sum($list);
?>
I wrote a program in PHP to find the largest prime factor. I think it is quite optimized, because it loads quite fast. But, there is a problem: it doesn't count the prime factors of very big numbers. Here is the program:
function is_even($s) {
$sk_sum = 0;
for($i = 1; $i <= $s; $i++) {
if($s % $i == 0) { $sk_sum++; }
}
if($sk_sum == 2) {
return true;
}
}
$x = 600851475143; $i = 2; //x is number
while($i <= $x) {
if($x % $i == 0) {
if(is_even($i)) {
$sk = $i; $x = $x / $i;
}
}
$i++;
}
echo $sk;
The largest non-overflowing integer in PHP is stored in the constant PHP_INT_MAX.
You won't be able to work with integers larger than this value in PHP.
To see all of PHP's predefined constants, just use:
<?php
echo '<pre>';
print_r(get_defined_constants());
echo '</pre>';
?>
PHP_INT_MAX probably has a value of 2,147,483,647.
To handle numbers of arbitrary precision in PHP, see either the GMP or BC Math PHP extensions.
You should read about Prime testing and Sieving.
In particular, you don't need to test whether each of your divisors is prime.
Something like the following would be faster.
while($i <= $x)
{
while ($x % $i == 0)
{
$sk = $i;
$x = $x / $i;
}
$i++;
}
You can also stop your outer loop when $i reaches sqrt($x), and if you haven't found a divisor yet then you know $x is prime.
Well, every language has it's own (while usually same) limitations, so if you exceed this php's limit, you can't get any higher. Max Integer is 9E18.