<< problem 131 - Prime cube partnership | Repunit nonfactors - problem 133 >> |
Problem 132: Large repunit factors
(see projecteuler.net/problem=132)
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k.
For example, R(10) = 1111111111 = 11 * 41 * 271 * 9091, and the sum of these prime factors is 9414.
Find the sum of the first forty prime factors of R(10^9).
My Algorithm
This problem was a tough one for me. I didn't realize that R(k) = dfrac{10^k - 1}{9} until I saw that equation on en.wikipedia.org/wiki/Repunit
The task is to find prime factors p such that R(k) == 0 mod p.
Which becomes (using the formula from Wikipedia):
dfrac{10^k - 1}{9} == 0 mod p
10^k - 1 == 0 mod 9p
10^k == 1 mod 9p
k is quite small, therefore a simplified version of my powmod
function (see my toolbox) can easily compute 10^k mod 9p
.
If it returns 1, then p
is added to sum
until 40 factors are found.
Note
My live tests allows you to enter different k and change the number of relevant prime factors, too (which is 40 in the original problem).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "10 4" | ./132
Output:
Note: the original problem's input 1000000000 40
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>
// return (base^exponent) % modulo
unsigned int powmod(unsigned int base, unsigned int exponent, unsigned int modulo)
{
unsigned int result = 1;
while (exponent > 0)
{
// fast exponentation:
// odd exponent ? a^b = a*a^(b-1)
if (exponent & 1)
result = (result * (unsigned long long)base) % modulo; // enforce 64-multiplication
// even exponent ? a^b = (a*a)^(b/2)
base = (base * (unsigned long long)base) % modulo; // enforce 64-multiplication
exponent >>= 1;
}
return result;
}
int main()
{
unsigned int numFactors = 40;
unsigned long long digits = 1000000000;
std::cin >> digits >> numFactors;
unsigned int sum = 0;
std::vector<unsigned int> primes = { 2 }; // two is never a prime factor of a repunit
for (unsigned int i = 3; numFactors > 0; i++)
{
bool isPrime = true;
// test against all prime numbers we have so far (in ascending order)
for (auto x : primes)
{
// prime is too large to be a divisor
if (x*x > i)
break;
// divisible => not prime
if (i % x == 0)
{
isPrime = false;
break;
}
}
// no prime
if (!isPrime)
continue;
primes.push_back(i);
// check for divisibility by 9*prime
auto modulo = 9 * i;
// remainder must be 1
auto remainder = powmod(10, digits, modulo);
if (remainder == 1)
{
sum += i;
numFactors--;
}
// don't analyze huge primes
// => not needed for the original problem, but avoids malicious input in live test
// => e.g. user asks for more primes than the repunit has
if (i > 1111111)
break;
}
std::cout << sum << std::endl;
return 0;
}
This solution contains 12 empty lines, 13 comments and 2 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
June 1, 2017 submitted solution
June 1, 2017 added comments
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Links
projecteuler.net/thread=132 - 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-132-large-repunit-factors/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p132.java (written by Nayuki)
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 131 - Prime cube partnership | Repunit nonfactors - problem 133 >> |