Problem 87: Prime power triples

(see projecteuler.net/problem=87)

The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28.
In fact, there are exactly four numbers below fifty that can be expressed in such a way:

28 = 2^2 + 2^3 + 2^4
33 = 3^2 + 2^3 + 2^4
49 = 5^2 + 2^3 + 2^4
47 = 2^2 + 3^3 + 2^4

How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?

Algorithm

A simple prime sieve is responsible to find all primes i < sqrt{50000000}.
Then three nested loops compute all sums of all combinations of such primes where a^2 + b^3 + c^4 < 50000000.
Be careful: b*b*b and c*c*c*c can easily exceed an unsigned int.

Those sums are sorted and duplicate sums are removed. Now we have a nice std::vector with all sums.
The original problem is solved now (just display sums.size()) but the Hackerrank problem is slightly tougher.

Modifications by HackerRank

In order to find the the number of sums which are below a certain input value, I search through the sorted container with std::upper_bound
and then compute the distance to the beginning of the container.
That's extremely fast (std::upper_bound most likely uses binary search) and easily processed thousands of test cases per second.

My code

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

#include <vector>
#include <algorithm>
#include <iostream>
 
int main()
{
const unsigned int MaxLimit = 100 * 1000 * 1000; // Hackerrank: 10^7 instead of 5*10^6
 
// prime sieve
std::vector<unsigned int> primes;
primes.push_back(2);
for (unsigned int i = 3; i*i < MaxLimit; i += 2)
{
bool isPrime = true;
 
// test against all prime numbers we have so far (in ascending order)
for (auto p : primes)
{
// next prime is too large to be a divisor ?
if (p*p > i)
break;
 
// divisible ? => not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
 
// yes, we have a prime number
if (isPrime)
primes.push_back(i);
}
 
// just three nested loops where I generate all sums
std::vector<unsigned int> sums;
for (auto a : primes)
for (auto b : primes)
for (auto c : primes)
{
auto a2 = a*a;
auto b3 = (unsigned long long)b*b*b;
auto c4 = (unsigned long long)c*c*c*c;
auto sum = a2 + b3 + c4;
// abort if too big
if (sum > MaxLimit)
break;
 
sums.push_back(sum);
}
 
// sort ascendingly
std::sort(sums.begin(), sums.end());
// a few sums occur twice, let's remove them !
auto last = std::unique(sums.begin(), sums.end());
 
// process test cases
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int limit = MaxLimit;
std::cin >> limit;
 
// find next sum which is bigger than the limit
auto pos = std::upper_bound(sums.begin(), last, limit);
// how many sums are inbetween 28 and limit ?
auto num = std::distance(sums.begin(), pos);
std::cout << num << std::endl;
}
 
return 0;
}

This solution contains 11 empty lines, 12 comments and 3 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 50" | ./87

Output:

(please click 'Go !')

Note: the original problem's input 50000000 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 0.16 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 20 MByte.

(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

March 12, 2017 submitted solution
May 5, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler087

My code solves 9 out of 9 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 20% (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.

Links

projecteuler.net/thread=87 - 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-87-sum-power/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p087.java (written by Nayuki)
Scala: github.com/samskivert/euler-scala/blob/master/Euler087.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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The 160 solved problems had an average difficulty of 21.8% at Project Euler and I scored 11,807 points (out of 13100) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage.
some names mentioned on this site may be trademarks of their respective owners.
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