# Find the nth prime number

It's a long time since I've written a Sieve of Eratosthenes (about 3 decades, I think, in ZX Basic). So I decided to revisit in the light of experience.

Rather than allocating a vector of unknown size to sieve values, I keep an ordered set of upcoming composite values which are updated as they are reached.

#include <map>

unsigned long find_nth_prime(unsigned long n)
{
if (!n) return 1;           // "0th prime"
if (!--n) return 2;         // first prime

// The map m contains one entry for each prime.  The key is the
// next time to update it, and the value is the amount to increment
// by each time it is reached.
auto m = std::map<unsigned long, unsigned long>();
// The overflow tests below are valid only for unsigned types.
for (auto i = 3ul;  true;  i += 2) {
if (i < 3ul)            // overflow test
throw std::overflow_error("prime too large");

const auto it = m.cbegin();
if (it == m.end() || it->first != i) {
// it's a prime
if (!--n)
return i;
if (i*i > i)        // overflow test
m.insert({i*i, 2*i});
} else {
// it's composite - advance the factor
auto next = it->first;
do {
next += it->second;
} while (next > i && !m.emplace(next, it->second).second);
m.erase(it);
}
}
}


And the test program (not really part of the review):

#include <iostream>
#include <string>

int main(int, char **argv)
{
std::cout.imbue(std::locale(""));
while (*++argv) {
try {
std::cout << find_nth_prime(std::stoul(*argv)) << std::endl;
} catch (std::exception&) {
std::cerr << "Invalid argument: " << *argv << std::endl;
}
}
}


Compiled with GCC:

g++ -std=c++17 -O3 -Wall -Wextra -Wpedantic


Whilst the performance is acceptable (about 1.6 seconds on my hardware to find the one-millionth prime - 15,485,863), I'm unsure whether a std::map is really the best structure for keeping track. As usual, any other critique of the code is welcome.

• Don't think 1 is a print number. – Martin York Feb 2 '17 at 17:17
• @Loki, no 1 isn't a prime number; I should have thrown invalid_argument there, I think. – Toby Speight Feb 2 '17 at 17:22

### Bug: Only works if longs are 64-bit

I ran your program on my 32-bit machine and got the wrong answer for the 1 millionth prime: 15498907 instead of 15485863. The problem is here:

        if (i*i > i)        // overflow test


If longs are 32 bits, then a number such as 0x10001 will pass this overflow test (because 0x10001 * 0x10001 = 0x20001 after overflowing). This eventually causes problems because overflowed values are inserted into the map. Actually, even if longs are 64-bit, you will still have a problem, but it will only happen after i reaches 0x100000001.

I fixed the problem by doing this:

    unsigned long max_i = std::sqrt(~0UL);
// ...
if (i <= max_i)      // overflow test


### Slow compared to vector based solution

I found your program to be a lot slower than a traditional vector based sieve. Here are timings for your program vs a traditional sieve that I wrote which used a vector<bool>:

Map    based,  1 millionth prime:  3.77 seconds
Map    based, 10 millionth prime: 47.91 seconds
Vector based,  1 millionth prime:  0.04 seconds
Vector based, 10 millionth prime:  0.93 seconds


### Sample vector based code

Since there was a comment asking about how the vector based code would work, here is the full program I used:

#include <iostream>
#include <string>
#include <cstdlib>
#include <cmath>
#include <cstdio>
#include <vector>

// This function uses the prime counting function approximation:
//
// n = x / ln(x)
//
// Where n is the number of primes below the number x.  Since we are trying
// to find the nth prime, we should solve for x here to determine an
// approximate value for the nth prime.  It turns out that for n >= 6, the
// value we solve for x will always be larger than the nth prime value.
//
// If the sieve size doesn't fit in an unsigned long, this function returns 0.
unsigned long findSieveSize(unsigned long n)
{
// For small n, the formula returns a value too low, so we can just
// hardcode the sieve size to 5 (5th prime is 11).
if (n < 6)
return 13;

// We can't find a prime that will exceed ~0UL.
if (n >= (~0UL / std::log(~0UL)))
return 0;

// Binary search for the right value.
unsigned long low  = n;
unsigned long high = ~0UL - 1;
do {
unsigned long mid   = low + (high - low) / 2;
double        guess = mid / std::log(mid);

if (guess > n)
high = (unsigned long) mid - 1;
else
low = (unsigned long) mid + 1;
} while (low < high);
return high + 1;
}

unsigned long find_nth_prime(unsigned long n)
{
if (!n) return 1;           // "0th prime"
if (!--n) return 2;         // first prime

unsigned long sieveSize = findSieveSize(n);
unsigned long count     = 0;
unsigned long max_i     = std::sqrt(sieveSize-1)+1;

if (sieveSize == 0)
return 0;

std::vector<bool> sieve(sieveSize);
for (unsigned long i = 3;  true;  i += 2) {
if (!sieve[i]) {
if (++count == n)
return i;
if (i >= max_i)
continue;
unsigned long j    = i*i;
unsigned long inc  = i+i;
unsigned long maxj = sieveSize - inc;
// This loop checks j before adding inc so that we can stop
// before j overflows.
do {
sieve[j] = true;
if (j >= maxj)
break;
j += inc;
} while (1);
}
}
return 0;
}

int main(int, char **argv)
{
std::cout.imbue(std::locale(""));
while (*++argv) {
try {
std::cout << find_nth_prime(atoi(*argv)) << std::endl;
} catch (std::exception&) {
std::cerr << "Invalid argument: " << *argv << std::endl;
}
}
}

• When you used std::vector<bool>, how did you know what size vector to allocate? – Toby Speight Feb 2 '17 at 21:45
• I used the prime counting formula from here, which is essentially $n = x / \ln {x}$. So I solved for $x$ given $n$. I was prepared to increase $n$ by some amount in order to account for the fact that the formula only approximates the right value, but I found that the value of $x$ I got was always already higher than the prime number I was looking for. – JS1 Feb 2 '17 at 23:15
• @TobySpeight I added the full program I used for the vector implementation, so you can see how I found the sieve size by doing a binary search for x based on the formula n = x / ln(x). – JS1 Feb 3 '17 at 0:18
• You might want to remove the parts you don't change (2 includes + main), so people don't search for nonexistent differences... – Deduplicator Feb 3 '17 at 0:43
• Thanks for the review - interesting to discover the performance hit of all those small allocations in the map is greater than any saving from not keeping the whole sieve in memory. And thanks for spotting that bug! (I'm normally able to use the GCC builtins for testing overflow, but wanted to post a portable program here). – Toby Speight Feb 3 '17 at 9:47

The treatment of the invalid argument can be improved:

if (!n) return 1;           // "0th prime"


Instead of returning a dummy value, it would be clearer to throw an exception:

if (!n) throw std::invalid_argument("n must be at least 1");


Also, the exception detail is lost when reporting:

std::cerr << "Invalid argument: " << *argv << std::endl;


We can print it if we name the caught exception:

std::cerr << "Invalid argument " << *argv << ": " << e.what() << std::endl;

• This answer is to summarize points made in comments or spotted after posting the question. – Toby Speight Feb 3 '17 at 9:41

Your code should express intent. Neither of these express intent. They look more like optimizations.

if (!n) return 1;           // "0th prime"
if (!--n) return 2;         // first prime


This expresses intent.

if (n == 0) return 1;           // "0th prime"
if (n == 1) return 2;           // first prime


Is this valid?

int main(int, char **argv)


This is valid:

int main(int argc, char *argv[])


Not sure this does what you think:

std::stoul(*argv)


Is argv[0] not the name of the application and argv[1] is the first user supplied argument.

Must admit I don't understand your implementation. So a decent description in comments about how that implements a sieve would definitely not be out of place.

• Naming argc serves only to introduce a compilation warning. I don't like warnings, and strive to fix them. You (@Loki) are right that I was lazy implementing main(), and it would be clearer with for (int i = 1; i < argc; ++i) f(argv[i]);. I've probably picked up some bad habits over at [code-golf.se]! – Toby Speight Feb 3 '17 at 9:40
• The suggested re-write of if (!n) return 1; if (!--n) return 2; looks nicer, yet changed functionality. n-- missing. – chux - Reinstate Monica Apr 10 '18 at 12:13
• @chux Don't know what I was thinking when I wrote that. 3.6.1.2 pretty much states the same in the 2nd half. Removed the original comment since it was partially incorrect and uneditable. – Snowhawk Apr 11 '18 at 3:06
• @Snowhawk Hmmm interesting - here I thought all this question actively was 2018 and not last year. – chux - Reinstate Monica Apr 11 '18 at 3:11

# Use a heap instead of a map

Standard Library provides std::priority_queue, which provides fast access to the highest value (which we can make the lowest with a suitable < operator). I was able to reduce runtime by using this class instead of std::map.

# Use a wheel to eliminate many composite numbers

A 2,3,5 wheel reduced runtime by a further 50% . A large part of this gain is by skipping over wheel-composite numbers when we increment positions; this greatly reduces updates to the container.

# Improved code

On my system, the modified program takes 0.6 seconds to find the millionth prime, compared with 1.6 seconds for the original (both using gcc -O3):

// Include the headers we'll need, and define the result type:

#include <algorithm>
#include <cmath>
#include <queue>
#include <stdexcept>
#include <tuple>

using Number = unsigned long;


// The wheel is just a table lookup:

static constexpr bool wheel_prime(Number i)
{
// mod-30 wheel eliminates multiples of 3 and 5
// incrementing by 2 eliminates even numbers
constexpr int wheel_size = 30;
constexpr bool wheel[2][wheel_size] = {
{
false,
false,
true, //  2
true, //  3
false,
true, //  5
false,
true, //  7
false,
false,
false,
true, //  11
false,
true, //  13
false,
false,
false,
true, //  17
false,
true, //  19
false,
false,
false,
true, //  23
false,
false,
false,
false,
false,
true, //  29
},
{
false,
true, //  1
false,
false,
false,
false,
false,
true, //  7
false,
false,
false,
true, //  11
false,
true, //  13
false,
false,
false,
true, //  17
false,
true, //  19
false,
false,
false,
true, //  23
false,
false,
false,
false,
false,
true, //  29
}
};
return wheel[i >= wheel_size][i % wheel_size];
}


// Use a "peg" class to represent a marker that moves along the sieve:

// Each prime has an associated "peg" that moves along the sieve.
class prime_peg
{
Number n; // current position
Number i; // the increment (2*p)

public:
explicit prime_peg(Number p)
: n{p*p},
i{p+p}
{}

operator Number() const { return n; }
prime_peg& operator++() { n += i; return *this; }

// To use in priority_queue, sort largest first
bool operator<(const prime_peg& other) const {
return std::tuple(n, i) > std::tuple(other.n, other.i);
}
};


// Set up the heap of pegs, and run the algorithm:

unsigned long find_nth_prime(unsigned long n)
{
// no need to insert any primes higher than this
static Number const max_peg = std::sqrt(~0UL);

// deal with small numbers
if (!n) throw std::invalid_argument("Requested zeroth prime");
if (!--n) return 2;         // the only even prime
if (!--n) return 3;         // wheel prime
if (!--n) return 5;         // wheel prime

// The queue m contains one peg for each prime.
auto m = std::priority_queue<prime_peg>{};

Number i = 7;               // first non-wheel prime
for (auto next = prime_peg{i};  i < ~0UL/2;  i+=2) {
if (!wheel_prime(i)) continue;
{
if (i < next) {
if (!--n) break;
if (i < max_peg) m.emplace(i);
}
}

for (;  next <= i;  next = m.top()) {
m.pop();
do { ++next; } while (!wheel_prime(next));
m.push(next);
}
}
return i;
}


// I've made some small improvements to the test code based on other reviews:

#include <iostream>
#include <string>

int main(int argc, char **argv)
{
std::cout.imbue(std::locale(""));
for (int i = 1;  i < argc;  ++i) {
try {
std::cout << find_nth_prime(std::stoul(argv[i])) << std::endl;
} catch (std::exception& e) {
std::cerr << "Invalid argument: " << argv[i]
<< ": " << e.what() << std::endl;
}
}
}