<< problem 6 - Sum square difference | Largest product in a series - problem 8 >> |

# 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?

# My 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 rarely 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

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

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort.

<< problem 6 - Sum square difference | Largest product in a series - problem 8 >> |