<< problem 34 - Digit factorials Double-base palindromes - problem 36 >>

Problem 35: Circular primes

The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.

There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.

How many circular primes are there below one million?

Algorithm

First, a a standard prime sieve finds all prime numbers up to our limit (1000000 by default) and keeps them in a std::set.

Then each prime x in std::set is rotated by one digit to the right:
1. get the right-most digit:
auto digit = rotated % 10;
2. move all digits by one digit to the right ("erasing" the right-most digit):
rotated /= 10;
3. prepend the right-most digit:
 rotated += digit * shift;
4. check whether rotated is part of our std::set, too
5. if rotated is equal to our initial value x then we checked all rotations

The only point of interest is shift which is a power of 10 such that 10^a = shift <= x <= 10^{a+1}.
E.g., if x = 3456 then shift = 1000.

Modifications by HackerRank

We have to find the sum of all such prime numbers, not their count.

Note

There are a few options to speed up the code:
1. All prime numbers are odd (except for 2): if x != 2 and any digit is even then this prime can't be circular.
2. We can simplify point 1 by noting that all single-digit primes are circular.

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 <iostream>
#include <set>

int main()
{
// highest number (1000000 in original problem)
unsigned int n;
std::cin >> n;

// precomputation: find all prime numbers up to n
std::set<unsigned int> primes;
primes.insert(2);
for (unsigned int i = 3; i <= n; i += 2)
{
bool isPrime = true;

// test against all prime numbers we have so far (in ascending order)
for (auto x : primes)
{
// divisible => not prime
if (i % x == 0)
{
isPrime = false;
break;
}

// prime is too large to be a divisor
if (x*x > i)
break;
}

// yes, we have a prime
if (isPrime)
primes.insert(i);
}

// now look at all primes
unsigned int sum = 0;
for (auto x : primes)
{
// move the right-most digit to the front of the number
// we need to know the "position" of the front-most digit:
// shift will be   1 for x =   1..9
// shift will be  10 for x =  10..99
// shift will be 100 for x = 100..999 and so on
unsigned int shift = 1;
while (x > shift * 10)
shift *= 10;

auto rotated = x;
do
{
// take right-most digit
auto digit = rotated % 10;
// remove it
rotated /= 10;
// and prepend it
rotated += digit * shift;

// rotated number not prime ?
if (primes.count(rotated) == 0)
break;
} while (rotated != x); // finished the circle ? (we have the initial number again)

// all rotations succeeded ?
#define ORIGINAL
#ifdef ORIGINAL
if (rotated == x)
sum++;
#else
if (rotated == x)
sum += x;
#endif
}

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


This solution contains 10 empty lines, 17 comments and 6 preprocessor commands.

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):

This is equivalent to
echo 100 | ./35

Output:

Note: the original problem's input 1000000 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)

Benchmark

The correct solution to the original Project Euler problem was found in 0.16 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 6 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 24, 2017 submitted solution

Hackerrank

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 never an option.

projecteuler.net/thread=35 - 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-35-circular-primes/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p035.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p035.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem35.c (written by eagletmt)
Javascript: github.com/dsernst/ProjectEuler/blob/master/35 Circular primes.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler035.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 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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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
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