<< problem 40 - Champernowne's constant | Coded triangle numbers - problem 42 >> |

# Problem 41: Pandigital prime

(see projecteuler.net/problem=41)

We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once.

For example, 2143 is a 4-digit pandigital and is also prime.

What is the largest n-digit pandigital prime that exists?

# Algorithm

The largest pandigital number is 987654321. In order to find out whether a number x <= 987654321 is prime, I precompute all primes up to sqrt{987654321} approx 31426.

These "small" prime numbers will be kept in `smallPrimes`

.

My second step is to generate all pandigital numbers: I create all permutations of the string `"123456789"`

and perform a simple primality test (using `smallPrimes`

).

Due to Hackerrank's variable number of digits, not only the 9-pandigitals numbers but also the 8-, 7-, 6-, ..., 2-pandigital numbers are checked, too.

The set `panPrimes`

will contain all 2-, ..., 9-pandigital primes after those two precomputation steps.

Each test case look ups the closest bigger pandigital prime (`upper_bound`

) and goes one step backwards.

## Note

When looking at the results I only saw 4- and 7-pandigital primes.

Modifying my loop in step 2 accordingly would provide a 10x speed-up.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <set>
#include <iostream>
#include <algorithm>
int main()
{
// precomputation step 1:
// find all primes below sqrt(987654321)
std::set<unsigned int> smallPrimes;
smallPrimes.insert(2);
for (unsigned int i = 3; i*i <= 987654321; i += 2)
{
bool isPrime = true;
for (auto p : smallPrimes)
{
// abort, no divisors possible
if (p*p > i)
break;
// divisor found ?
if (i % p == 0)
{
isPrime = false;
break;
}
}
// found a prime number
if (isPrime)
smallPrimes.insert(i);
}
// precomputation step 2:
// generate all permutations of the strings "12", "123", "1234", ..., "123456789"
// and test whether they are prime
std::set<unsigned int> panPrimes;
for (unsigned int digits = 2; digits <= 9; digits++)
{
std::string strNumber = "123456789";
// reduce number of digits
strNumber.erase(digits);
do
{
unsigned int number = std::stoi(strNumber);
// test whether pandigital number is prime
bool isPrime = true;
for (auto p : smallPrimes)
{
// abort, no divisors possible
if (p*p > number)
break;
// divisor found ?
if (number % p == 0)
{
isPrime = false;
break;
}
}
// found a pandigital prime ?
if (isPrime)
panPrimes.insert(number);
} while (std::next_permutation(strNumber.begin(), strNumber.end()));
}
// process input
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int limit;
std::cin >> limit;
// find next larger pandigital prime number
auto i = panPrimes.upper_bound(limit);
// smaller than the smallest pandigital prime ?
if (i == panPrimes.begin())
{
std::cout << "-1" << std::endl;
continue;
}
// upper_bound() goes one step too far
i--;
// and print it
std::cout << *i << std::endl;
}
return 0;
}

This solution contains 13 empty lines, 18 comments and 3 preprocessor commands.

# Interactive test

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

This is equivalent to`echo "1 2500" | ./41`

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 0.03 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(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 25, 2017 submitted solution

April 19, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler041

My code solves **5** out of **5** 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=41 - **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-41-pandigital-prime/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p041.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p041.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem41.cc (written by eagletmt)

Javascript: github.com/dsernst/ProjectEuler/blob/master/41 Pandigital prime.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler041.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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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

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