<< problem 345 - Matrix Sum | Largest integer divisible by two primes - problem 347 >> |
Problem 346: Strong Repunits
(see projecteuler.net/problem=346)
The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6
(i.e. 7_10 = 11_6 = 111_2). In other words, 7 is a repunit in at least two bases b > 1.
We shall call a positive integer with this property a strong repunit.
It can be verified that there are 8 strong repunits below 50: {1,7,13,15,21,31,40,43}.
Furthermore, the sum of all strong repunits below 1000 equals 15864.
Find the sum of all strong repunits below 10^12.
My Algorithm
Each number x is a repunit in base x-1, more specific it's 11_{x-1} = (x-1)^1 + (x-1)^0 = x-1 + 1 = x.
A repunit in base 10 is 10^k + 10^{k-1} + 10^{k-2} + ... + 10^2 + 10^1 + 10^0.
The same pattern surfaces for any base b: b^k + b^{k-1} + b^{k-2} + ... + b^2 + b^1 + b^0
If I chase this sequence for each base b up to the limit 10^12 then I will find all strong repunits.
The only problem is that some numbers are strong repunits in more than two bases.
Those numbers are easily filtered by sorting and removing consecutive duplicates.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 1000 | ./346
Output:
Note: the original problem's input 1000000000000
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>
#include <algorithm>
int main()
{
// search limit
auto limit = 1000000000000ULL;
std::cin >> limit;
// 1 is a strong unit in all bases
std::vector<unsigned long long> found = { 1 };
// analyze other bases
for (unsigned long long base = 2; base*base < limit; base++)
{
auto current = 1 + base + base * base;
auto powers = base * base;
while (current < limit)
{
found.push_back(current);
powers *= base;
current += powers;
}
}
// remove duplicates
std::sort(found.begin(), found.end());
auto last = std::unique(found.begin(), found.end());
found.erase(last, found.end());
// display sum of all elements
unsigned long long sum = 0;
for (auto x : found)
sum += x;
std::cout << sum << std::endl;
return 0;
}
This solution contains 5 empty lines, 5 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.14 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 11 MByte.
(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
July 9, 2017 submitted solution
July 9, 2017 added comments
Difficulty
Project Euler ranks this problem at 15% (out of 100%).
Links
projecteuler.net/thread=346 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/frrad/project-euler/blob/master/python/Problem346.py (written by Frederick Robinson)
Python github.com/Meng-Gen/ProjectEuler/blob/master/346.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p346.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/346.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/346.cpp (written by Yuping Luo)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p346.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem346.java (written by Magnus Solheim Thrap)
Perl github.com/gustafe/projecteuler/blob/master/346-Strong-Repunits.pl (written by Gustaf Erikson)
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 345 - Matrix Sum | Largest integer divisible by two primes - problem 347 >> |