<< problem 68 - Magic 5-gon ring | Totient permutation - problem 70 >> |
Problem 69: Totient maximum
(see projecteuler.net/problem=69)
Euler's Totient function, phi(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n.
For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, phi(9)=6.
nRelatively Primephi(n)n/phi(n)
2112
31,221.5
41,322
51,2,3,441.25
61,523
71,2,3,4,5,661.1666...
81,3,5,742
91,2,4,5,7,861.5
101,3,7,942.5
It can be seen that n=6 produces a maximum n/phi(n) for n <= 10.
Find the value of n <= 1000000 for which n/phi(n) is a maximum.
My Algorithm
I have no formal proof yet (it's too late - time to go to bed !):
in my first brute-force attempt I observed that the "best number" is the product of all primes
best = 2 * 3 * 5 * 7 * 11 * 13 * ... where best < 1000000.
Simple tests showed that all primes from 2 to 57 are sufficient.
Modifications by HackerRank
The test best * nextPrime >= limit
might overflow.
The same result can be achieved this way:
best * nextPrime >= limit
best >= dfrac{limit}{nextPrime}
All variables are integers and thus rounding comes into play.
The correct formula is:
best >= dfrac{limit}{nextPrime} + dfrac{nextPrime - 1}{nextPrime}
best >= dfrac{limit + nextPrime - 1}{nextPrime}
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 10" | ./69
Output:
(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>
int main()
{
// enough primes for this problem
const unsigned int primes[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 57 };
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long limit;
std::cin >> limit;
// multiply until we hit our limit
unsigned long long bestPos = 1;
for (auto p : primes)
{
// continue until bestPos reaches or exceeds our input value
//__int128 next = bestPos * p;
//if (next >= limit)
// break;
// same code as before but more portable:
if (bestPos >= (limit + p - 1) / p)
break;
bestPos *= p;
}
std::cout << bestPos << std::endl;
}
return 0;
}
This solution contains 5 empty lines, 7 comments and 1 preprocessor command.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 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
March 1, 2017 submitted solution
April 26, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler069
My code solves 8 out of 8 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 10% (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 rarely an option.
Links
projecteuler.net/thread=69 - 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-69-find-the-value-of-n-≤-1000000-for-which-nφn-is-a-maximum/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p069.java (written by Nayuki)
Go github.com/frrad/project-euler/blob/master/golang/Problem069.go (written by Frederick Robinson)
Scala github.com/samskivert/euler-scala/blob/master/Euler069.scala (written by Michael Bayne)
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 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 68 - Magic 5-gon ring | Totient permutation - problem 70 >> |