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 #ifdefs 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.

Number of test cases (1-5):

Input data (separated by spaces or newlines):
Note: This live test requires Hackerrank-style input: enter a number and the program returns the number of ways it can be represented as the sum of a prime and twice a sqaure

This is equivalent to
echo "1 33" | ./46

Output:

(please click 'Go !')

(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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
The 126 solved problems had an average difficulty of 16.0% at Project Euler and I scored 11,074 points (out of 12500) at Hackerrank's Project Euler+.
more about me can be found on my homepage.
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