<< problem 57 - Square root convergents | XOR decryption - problem 59 >> |
Problem 58: Spiral primes
(see projecteuler.net/problem=58)
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that
8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of frac{8}{13} approx 62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed.
If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?
My Algorithm
My solution consists almost entirely of "old code":
- the Miller-Rabin primality test from problem 50 ...
- which in turn requires modular arithmetic from problem 48
Modifications by HackerRank
My portable mulmod
was a little bit too slow for Hackerrank, therefore I added the "GCC"-Hack:
the GCC-extension __int128
simplifies mulmod
to a one-line function (which is much faster, too).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 60 | ./58
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. Or just jump to my GitHub repository.
#include <iostream>
// return (a*b) % modulo
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
{
#ifdef __GNUC__
// use GCC's optimized 128 bit code
return ((unsigned __int128)a * b) % modulo;
#endif
// (a * b) % modulo = (a % modulo) * (b % modulo) % modulo
a %= modulo;
b %= modulo;
// fast path
if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
return (a * b) % modulo;
// we might encounter overflows (slow path)
// the number of loops depends on b, therefore try to minimize b
if (b > a)
std::swap(a, b);
// bitwise multiplication
unsigned long long result = 0;
while (a > 0 && b > 0)
{
// b is odd ? a*b = a + a*(b-1)
if (b & 1)
{
result += a;
if (result >= modulo)
result -= modulo;
// skip b-- because the bit-shift at the end will remove the lowest bit anyway
}
// b is even ? a*b = (2*a)*(b/2)
a <<= 1;
if (a >= modulo)
a -= modulo;
// next bit
b >>= 1;
}
return result;
}
// return (base^exponent) % modulo
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)
{
unsigned long long result = 1;
while (exponent > 0)
{
// fast exponentation:
// odd exponent ? a^b = a*a^(b-1)
if (exponent & 1)
result = mulmod(result, base, modulo);
// even exponent ? a^b = (a*a)^(b/2)
base = mulmod(base, base, modulo);
exponent >>= 1;
}
return result;
}
// Miller-Rabin-test
bool isPrime(unsigned long long p)
{
// IMPORTANT: requires mulmod(a, b, modulo) and powmod(base, exponent, modulo)
// some code from https://ronzii.wordpress.com/2012/03/04/miller-rabin-primality-test/
// with optimizations from http://ceur-ws.org/Vol-1326/020-Forisek.pdf
// good bases can be found at http://miller-rabin.appspot.com/
// trivial cases
const unsigned int bitmaskPrimes2to31 = (1 << 2) | (1 << 3) | (1 << 5) | (1 << 7) |
(1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) |
(1 << 23) | (1 << 29); // = 0x208A28Ac
if (p < 31)
return (bitmaskPrimes2to31 & (1 << p)) != 0;
if (p % 2 == 0 || p % 3 == 0 || p % 5 == 0 || p % 7 == 0 || // divisible by a small prime
p % 11 == 0 || p % 13 == 0 || p % 17 == 0)
return false;
if (p < 17*19) // we filtered all composite numbers < 17*19, all others below 17*19 must be prime
return true;
// test p against those numbers ("witnesses")
// good bases can be found at http://miller-rabin.appspot.com/
const unsigned int STOP = 0;
const unsigned int TestAgainst1[] = { 377687, STOP };
const unsigned int TestAgainst2[] = { 31, 73, STOP };
const unsigned int TestAgainst3[] = { 2, 7, 61, STOP };
// first three sequences are good up to 2^32
const unsigned int TestAgainst4[] = { 2, 13, 23, 1662803, STOP };
const unsigned int TestAgainst7[] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022, STOP };
// good up to 2^64
const unsigned int* testAgainst = TestAgainst7;
// use less tests if feasible
if (p < 5329)
testAgainst = TestAgainst1;
else if (p < 9080191)
testAgainst = TestAgainst2;
else if (p < 4759123141ULL)
testAgainst = TestAgainst3;
else if (p < 1122004669633ULL)
testAgainst = TestAgainst4;
// find p - 1 = d * 2^j
auto d = p - 1;
d >>= 1;
unsigned int shift = 0;
while ((d & 1) == 0)
{
shift++;
d >>= 1;
}
// test p against all bases
do
{
auto x = powmod(*testAgainst++, d, p);
// is test^d % p == 1 or -1 ?
if (x == 1 || x == p - 1)
continue;
// now either prime or a strong pseudo-prime
// check test^(d*2^r) for 0 <= r < shift
bool maybePrime = false;
for (unsigned int r = 0; r < shift; r++)
{
// x = x^2 % p
// (initial x was test^d)
x = powmod(x, 2, p);
// x % p == 1 => not prime
if (x == 1)
return false;
// x % p == -1 => prime or an even stronger pseudo-prime
if (x == p - 1)
{
// next iteration
maybePrime = true;
break;
}
}
// not prime
if (!maybePrime)
return false;
} while (*testAgainst != STOP);
// prime
return true;
}
int main()
{
unsigned int percentage = 10;
std::cin >> percentage;
// the lower right diagonal contains only squares
unsigned long long numPrimes = 0;
unsigned long long sideLength = 1;
unsigned long long diagonals = 1;
do
{
sideLength += 2;
diagonals += 4;
unsigned long long lowerRight = sideLength * sideLength;
unsigned long long lowerLeft = lowerRight - (sideLength - 1);
unsigned long long upperLeft = lowerLeft - (sideLength - 1);
unsigned long long upperRight = upperLeft - (sideLength - 1);
// no need to test lowerRight since it's a square and never prime
if (isPrime(lowerLeft) != 0)
numPrimes++;
if (isPrime(upperLeft) != 0)
numPrimes++;
if (isPrime(upperRight) != 0)
numPrimes++;
} while (numPrimes * 100 / diagonals >= percentage);
std::cout << sideLength << std::endl;
return 0;
}
This solution contains 28 empty lines, 40 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.02 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(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 28, 2017 submitted solution
April 24, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler058
My code solves 9 out of 9 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.
Similar problems at Project Euler
Problem 50: Consecutive prime sum
Problem 60: Prime pair sets
Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.
Links
projecteuler.net/thread=58 - 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-58-primes-diagonals-spiral/ (written by Kristian Edlund)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-058.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p058.py (written by Nayuki)
Python github.com/sefakilic/euler/blob/master/python/euler058.py (written by Sefa Kilic)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/058.cpp (written by Haochen Liu)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/58.cc (written by Meng-Gen Tsai)
Java github.com/dcrousso/ProjectEuler/blob/master/PE058.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p058.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem58.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem058.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p058.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/058.nb
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_058.clj (written by Guillaume Nargeot)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler058.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler058.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/058-Spiral-primes.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p058.rs
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 |
I stopped working on Project Euler problems around the time they released 617.
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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 57 - Square root convergents | XOR decryption - problem 59 >> |