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Problem 46: Goldbach's other conjecture
(see projecteuler.net/problem=46)
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
9 = 7 + 2 * 1^2
15 = 7 + 2 * 2^2
21 = 3 + 2 * 3^2
25 = 7 + 2 * 3^2
27 = 19 + 2 * 2^2
33 = 31 + 2 * 1^2
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
My Algorithm
A standard prime sieve quickly finds all primes up to 500000.
For all odd numbers i my program generate all squares j^2 < i.
If no j exists such that i - j^2 is prime then Goldbach's other conjecture is refuted.
Modifications by HackerRank
Again, the Hackerrank problem is significantly different from the original Project Euler problem:
we have to find all ways to represent it as a sum of a prime number and twice a square.
My program generates all squares j^2 < i and count how often i - j^2 is prime.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This live test is based on the Hackerrank problem.
This is equivalent toecho "1 33" | ./46
Output:
(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.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <set>
#include <iostream>
int main()
{
const unsigned int MaxPrime = 500000;
// find all primes up to 500000
std::set<unsigned int> primes;
primes.insert(2);
for (unsigned int i = 3; i < MaxPrime; 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.insert(i);
}
//#define ORIGINAL
#ifdef ORIGINAL
// start at 9 (smallest odd number which is not prime)
for (unsigned int i = 9; i <= MaxPrime; i += 2)
{
// only composite numbers
if (primes.count(i) != 0)
continue;
bool refuteConjecture = true;
// try all squares
for (unsigned int j = 1; 2*j*j < i; j++)
{
auto check = i - 2*j*j;
// found a combination, conjecture is still valid
if (primes.count(check) != 0)
{
refuteConjecture = false;
break;
}
}
// conjecture refuted !
if (refuteConjecture)
{
std::cout << i << std::endl;
break;
}
}
#else
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int i;
std::cin >> i;
// try all squares
unsigned int solutions = 0;
for (unsigned int j = 1; 2*j*j < i; j++)
{
// check whether i - j^2 is prime
unsigned int check = i - 2*j*j;
// yes, found another combination
if (primes.count(check) != 0)
solutions++;
}
std::cout << solutions << std::endl;
}
#endif
return 0;
}
This solution contains 11 empty lines, 14 comments and 5 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.07 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 4 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
February 27, 2017 submitted solution
April 19, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler046
My code solves 6 out of 6 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.
Links
projecteuler.net/thread=46 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-46-odd-number-prime-square/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem46.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p046.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/46 Goldbach's other conjecture.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem046.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p046.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler046.scala (written by Michael Bayne)
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 |
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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 45 - Triangular, pentagonal, and hexagonal | Distinct primes factors - problem 47 >> |