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:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

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

Output:

(please click 'Go !')

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 solved 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 already 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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
more about me can be found on my homepage.
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