<< problem 132 - Large repunit factors | Prime pair connection - problem 134 >> |
Problem 133: Repunit nonfactors
(see projecteuler.net/problem=133)
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(6) = 111111.
Let us consider repunits of the form R(10^n).
Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17.
Yet there is no value of n for which R(10^n) will divide by 19. In fact, it is remarkable that 11, 17, 41, and 73 are the only four primes below one-hundred that can be a factor of R(10^n).
Find the sum of all the primes below one-hundred thousand that will never be a factor of R(10n).
My Algorithm
After solving problem 132 I played around with the code and just got lucky:
I tried looping over several repunits but it turned out it is sufficient to test only one really, really large repunit.
Unfortunately, I cannot give any satisfying reasoning. I saw some proves on other websites, though.
In order to work with such "really, really large repunit" I had to swap my powmod
code from problem 132 with code that can handle 64 bit values.
It's part of my toolbox, 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 "1 100" | ./133
Output:
Note: the original problem's input 100000
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 (a*b) % modulo
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
{
// (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;
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;
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;
}
int main()
{
// a large exponent for powmod => 10^(10^19)
const unsigned long long digits = 10000000000000000000ULL;
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int maxPrime = 100000;
std::cin >> maxPrime;
unsigned long long sum = 0;
std::vector<unsigned int> primes;
for (unsigned int i = 2; i < maxPrime; 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 not be 1
auto remainder = powmod(10, digits, modulo);
if (remainder != 1)
sum += i;
}
std::cout << sum << std::endl;
}
return 0;
}
This solution contains 19 empty lines, 21 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 2, 2017 submitted solution
June 2, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler133
My code solves 9 out of 11 test cases (score: 80%)
I failed 0 test cases due to wrong answers and 2 because of timeouts
Difficulty
Project Euler ranks this problem at 50% (out of 100%).
Hackerrank describes this problem as medium.
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.
Links
projecteuler.net/thread=133 - 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-133-repunit-nonfactors/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p133.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 a new 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 132 - Large repunit factors | Prime pair connection - problem 134 >> |