<< problem 516 - 5-smooth totients | First Sort I - problem 523 >> |
Problem 518: Prime triples and geometric sequences
(see projecteuler.net/problem=518)
Let S(n) = sum{a+b+c} over all triples (a,b,c) such that:
- a, b, and c are prime numbers.
- a < b < c < n.
- a+1, b+1, and c+1 form a geometric sequence.
(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17),
(7, 23, 71), (11, 23, 47), (17, 23, 31), (17, 41, 97), (31, 47, 71), (71, 83, 97)
Find S(10^8).
My Algorithm
This problem is somehow similar to problem 141:
I learnt from the 141 forum that a geometric progression can be represented as kyy, kxy, kxx where x > y and gcd(x, y) = 1
I wrote three nested loops for k, x and y which check whether kxx - 1, kxy - 1 and kyy - 1 are prime numbers.
My code contains a few optimizations:
- the loop nesting order x, k, y is roughly 30% faster than the more obvious x, y, k
- I deferred the gcd() test because of its slow modulo computations
- ... but early on do a quick bit test whether x and y are both even → because that implies gcd(x, y) > 1
Note
You will find a little "debugging helper", too: I implemented a simple brute-force algorithm (see bruteForce
) which helped me
to locate two bugs I had in my first version of count
(which is the final/fast version).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 100 | ./518
Output:
Note: the original problem's input 100000000
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)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#include <iostream>
#include <vector>
#include <cmath>
// ---------- standard prime sieve from my toolbox ----------
// odd prime numbers are marked as "true" in a bitvector
std::vector<bool> sieve;
// return true, if x is a prime number
bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size
void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = (size >> 1) + 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2*i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3*i+1;
while (current < half)
{
sieve[current] = false;
current += 2*i+1;
}
}
}
// greatest common divisor
template <typename T>
T gcd(T a, T b)
{
while (a != 0)
{
T c = a;
a = b % a;
b = c;
}
return b;
}
// ---------- problem specific code ----------
// slow approach
unsigned long long bruteForce(unsigned int limit)
{
unsigned long long sum = 0;
// look for geometric progression a+1, b+1, c+1 where all three are prime numbers
// iterate over all prime numbers a and c where a < c
for (unsigned int a = 2; a < limit; a++)
{
if (!isPrime(a))
continue;
for (auto c = a + 4; c < limit; c++)
{
if (!isPrime(c))
continue;
// (b+1)^2 = (a+1) * (c+1)
auto b2 = (a + 1) * (c + 1);
unsigned int b = sqrt(b2);
if (b * b != b2)
continue;
b--;
// of course: b must be prime, too
if (!isPrime(b))
continue;
// found one more solution
std::cout << a << " " << b << " " << c << std::endl;
sum += a + b + c;
}
}
return sum;
}
// find triples k*x*x, k*x*y, k*y*y whose predecessors are prime
unsigned long long count(unsigned int limit)
{
unsigned long long sum = 0;
// I tried a few variations and the following order of loops turned out to be the fastest:
// x => k => y
// the more obvious order x => y => k is about 30% slower on my computer
for (unsigned int x = 2; x*x < limit; x++)
for (unsigned int k = 1; k*x*x < limit; k++)
{
auto a = k*x*x - 1;
if (!isPrime(a))
continue;
for (unsigned int y = 1; y < x; y++)
{
// performance tweak: if both x and y are even then the gcd() test will fail
if ((x & 1) == 0 && (y & 1) == 0)
y++;
auto b = k*x*y - 1;
auto c = k*y*y - 1;
if (!isPrime(b))
continue;
if (!isPrime(c))
continue;
// avoid duplicate solutions
if (gcd(x, y) > 1)
continue;
// note: I delayed this gcd() for performance reasons
// it's more "expensive" to check right after the beginning of the y-loop
// found one more solution
sum += a + b + c;
}
}
return sum;
}
int main()
{
unsigned int limit = 100000000;
std::cin >> limit;
fillSieve(limit);
//std::cout << bruteForce(limit) << std::endl;
std::cout << count(limit) << std::endl;
return 0;
}
This solution contains 26 empty lines, 30 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.9 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 8 MByte.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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
September 2, 2017 submitted solution
September 2, 2017 added comments
Difficulty
Project Euler ranks this problem at 20% (out of 100%).
Links
projecteuler.net/thread=518 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/HaochenLiu/My-Project-Euler/blob/master/518.py (written by Haochen Liu)
Python github.com/Meng-Gen/ProjectEuler/blob/master/518.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p518.py (written by Nayuki)
C++ github.com/evilmucedin/project-euler/blob/master/euler518/518.cpp (written by Den Raskovalov)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/518.cpp (written by Haochen Liu)
C++ github.com/roosephu/project-euler/blob/master/518.cpp (written by Yuping Luo)
C github.com/HaochenLiu/My-Project-Euler/blob/master/518_mt.c (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p518.java (written by Nayuki)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
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 516 - 5-smooth totients | First Sort I - problem 523 >> |