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# Problem 7: 10001st prime

(see projecteuler.net/problem=7)

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10001st prime number?

# Algorithm

A prime number is an integer number p>=2 that can only be divided by 1 and by itself (p).

2 is the smallest prime number and the only even prime number, too (all other prime numbers are odd).

Each number x can be split into its prime factors, that means we check for all primes p<x whether x \mod p == 0.

If that test fails for all those primes, then x is a prime number and can be added to our `std::vector`

.

## Alternative Approaches

Take a look at my toolbox for other prime sieves or even precomputed lookup tables.

Wikipedia lists a few faster algorithms (en.wikipedia.org/wiki/Prime_number), too.

On my website create.stephan-brumme.com/eratosthenes/ you can find parallelized code that computes

all 50847534 prime numbers below 1 billion in less than a second.

## Note

Actually we can abort the loop if p>=sqrt{x} (which is p^2>=x) to speed up the program.

And since all primes are odd - *except for 2* - I simply add 2 to the list of primes and then scan

only odd numbers, beginning with 3 (and increment 2).

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
int main()
{
// compute the first 10001 primes
std::vector<unsigned int> primes;
primes.reserve(10001);
primes.push_back(2);
for (unsigned int x = 3; primes.size() <= 10000; x += 2)
{
bool isPrime = true;
for (auto p : primes)
{
// found a divisor ? => abort
if (x % p == 0)
{
isPrime = false;
break;
}
// no larger prime factors possible ?
if (p*p > x)
break;
}
// yes, we have a new prime
if (isPrime)
primes.push_back(x);
}
// processing all test cases is now just a plain lookup
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// just look up the x-th prime
// with a little twist: vector's index is zero-based, therefore "off by one"
x--;
if (x < primes.size())
std::cout << primes[x] << std::endl;
else
std::cout << "ERROR" << std::endl;
}
return 0;
}

In order to run my code, execute`echo "1 10001" | ./euler-0007`

(input format usually follows Hackerrank's requirements)

This solution contains 5 empty lines, 7 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 6" | ./7`

Output:

*Note:* the original problem's input `10001`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

February 23, 2017 submitted solution

March 28, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler007

My code solves **5** out of **5** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **easy**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Similar problems at Project Euler

Problem 10: Summation of primes

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=7 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-problem-7/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p007.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p007.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p007.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem7.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem007.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/7 10001st prime.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler007.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

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