<< problem 45 - Triangular, pentagonal, and hexagonal | Distinct primes factors - problem 47 >> |

# 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?

# 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.

# 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.

# 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 to`echo "1 33" | ./46`

Output:

*(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 **0.07** seconds on a 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=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 27, 2017 submitted solution

April 19, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler046

My code solved **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 never 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:

Python: www.mathblog.dk/project-euler-46-odd-number-prime-square/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p046.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p046.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem46.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem046.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/46 Goldbach's other conjecture.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler046.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:*

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