<< problem 227 - The Chase Fibonacci Words - problem 230 >>

# Problem 229: Four Representations using Squares

Consider the number 3600. It is very special, because
3600 = 48^2 + 36^2
3600 = 20^2 + 2 * 40^2
3600 = 30^2 + 3 * 30^2
3600 = 45^2 + 7 * 15^2

Similarly, we find that 88201 = 99^2 + 280^2 = 287^2 + 2 * 54^2 = 283^2 + 3 * 52^2 = 197^2 + 7 * 84^2.

In 1747, Euler proved which numbers are representable as a sum of two squares.
We are interested in the numbers n which admit representations of all of the following four types:
n = a_1^2 + b_1^2
n = a_2^2 + 2 b_2^2
n = a_3^2 + 3 b_3^2
n = a_7^2 + 7 b_7^2,
where the a_k and b_k are positive integers.

There are 75373 such numbers that do not exceed 10^7.
How many such numbers are there that do not exceed 2 * 10^9?

# My Algorithm

My program works similar to a prime sieve:
Two nested loops generate all a and b and then mark all x = a^2 + k * b^2 (where k \in { 1,2,3,7 }).
It's possible that there are multiple combinations of a and b for the same k.
Therefore I use four bits because there are four different k.

The naive approach needs 2 GByte RAM. A substantial part of my code is dedicated to process all x in a rolling fashion:
Iterate over all a and then over all b but restrict b in such a way that x = a^2 + k * b^2 is in a certain range.
My current design processes 1 million x at once - which requires 1 MByte RAM and fits nicely in my CPU L2 cache.
The four arrays b1, b2, b3 and b7 store the minimum b for each a for the current slice / range.

## Alternative Approaches

Some people came up with smart insights regarding special properties of the matching numbers modulo 168.
Apparantly you can solve the problem in 1 second. Mine needs about 11 seconds.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):
Note: Enter the upper search limit

This is equivalent to
echo 10000000 | ./229

Output:

Note: the original problem's input 2000000000 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.

       #include <iostream>
#include <vector>
#include <cmath>

const unsigned int SliceSize = 1000*1000;

// bitmasks of for a^2 + k * b^2
const unsigned char One   = 1 << 0; // k = 1
const unsigned char Two   = 1 << 1; // k = 2
const unsigned char Three = 1 << 2; // k = 3
const unsigned char Seven = 1 << 3; // k = 7
// every matching number must have all four bits set
const unsigned char All   = One | Two | Three | Seven;

int main()
{
unsigned int limit = 2*1000*1000*1000;
std::cin >> limit;

// how many numbers match the criteria
unsigned int count = 0;

// contains bitmasks for the current slice
std::vector<unsigned char> used(SliceSize, 0);

// store start values of b for each factor k=1,2,3,7
unsigned int maxA = sqrt(limit);
std::vector<unsigned int> b1(maxA + 1, 1);
std::vector<unsigned int> b2(maxA + 1, 1);
std::vector<unsigned int> b3(maxA + 1, 1);
std::vector<unsigned int> b7(maxA + 1, 1);

// split into slices
unsigned int from = 0;
while (from < limit)
{
// size of current slice
unsigned int to = from + SliceSize;
if (to > limit)
to = limit;

// process all a and b where a^2+kb^2 is within the current slice
for (unsigned int a = 1; a*a + b1[a]*b1[a] < to; a++)
{
// a^2 + b^2
unsigned int b = b1[a];
for (; a*a +   b*b < to; b++)
used[a*a +   b*b - from] |= One;
b1[a] = b;

// a^2 + 2b^2
b = b2[a];
for (; a*a + 2*b*b < to; b++)
used[a*a + 2*b*b - from] |= Two;
b2[a] = b;

// a^2 + 3b^2
b = b3[a];
for (; a*a + 3*b*b < to; b++)
used[a*a + 3*b*b - from] |= Three;
b3[a] = b;

// a^2 + 7b^2
b = b7[a];
for (; a*a + 7*b*b < to; b++)
used[a*a + 7*b*b - from] |= Seven;
b7[a] = b;
}

// count all matching numbers
for (auto& x : used)
{
if (x == All)
count++;

// reset for next iteration
x = 0;
}

from = to;
}

std::cout << count << std::endl;
return 0;
}


This solution contains 15 empty lines, 14 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 10.5 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

June 21, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 70% (out of 100%).

# 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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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
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.

 << problem 227 - The Chase Fibonacci Words - problem 230 >>
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