Project Euler is a series of challenging mathematical/computer programming problems.
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Linear Combinations of Irrational Numbers: An Analysis on Architecture
Under what condition(s) is
$$ k_1\omega_1+\cdots + k_n\omega_n=c,$$
where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$?
I'm essentially trying to show that this is the case only so ...
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3answers
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Largest prime factor of 600851475143 [duplicate]
I'm trying to use a program to find the largest prime factor of 600851475143.
This is for Project Euler here: http://projecteuler.net/problem=3
I first attempted this with the code that goes through ...
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1answer
197 views
How to find the smallest number with just $0$ and $1$ which is divided by a given number?
Every positive integer divide some number whose representation (base $10$) contains only zeroes and ones. One can easily prove that using pigeonhole principle.
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Find the sum of all the multiples of 3 or 5 of 1000000000 [closed]
Find the sum of all the multiples of 3 or 5 limit $1<n<100000000$
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1answer
70 views
taking the log of $a^b$ (Project Euler problem 29)
I've been stuck on Project Euler problem 29 and thus asked a friend who solved it how to do it.
What he basically did was for each power was: $\left(\frac{\log_{10}(a)}{\log_{10}(2)}\right)\cdot b$ ...
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2answers
61 views
Longest antichain of divisors
I Need to find a way to calculate the length of the longest antichain of divisors of a number N (example 720 - 6, or 1450 - 4), with divisibility as operation.
Is there a universally applicable way to ...
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1answer
251 views
Project Euler $420$ [closed]
So the question is:
We define $F(N)$ as the number of the $2\times 2$ positive integer matrices which have a trace less than $N$ and which can be expressed as a square of a positive integer matrix ...
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123 views
Pattern for extracting euler angles from a rotation matrix
There must be some kind of pattern when converting from one euler angle sequence to another, for example, if I were to convert from ZXY to ...
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2answers
67 views
Project Euler Problem 65
I am working on solving Project Euler problem #65 and run upon the following statement:
What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , ...
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3answers
67 views
Find smallest multiple of specific set of numbers
I was trying to solve the 5th problem on project-euler.net, wich is finding the smallest number wich was multiple of each number in a specific set, in this case, $[1 ... 20]$. First I thought of was ...
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1answer
38 views
quick approximation for largest fibonacci under a limit?
I asked in a previous post about finding a closed form for: $$\sum_{i=0}^{n}F_{3i}$$ which is the sum of the even fibs less than or equal to the nth even fib. the great answersI got showed me a very ...
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3answers
147 views
Closed form for the sum of even fibonacci numbers?
I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for
$$
\sum_{i=1}^n F_{3i}
$$
...
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1answer
61 views
How to get Euler angles with respect to initial Euler angle
I have a sensor which gives me Euler angles (roll,pitch,yaw). There is a baseline value of Euler angle (assume it is $5,10,15$) at the beginning.I want to calibrate from this baseline values all ...
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2answers
65 views
Combinatorical meaning of an identity involving factorials [duplicate]
While solving (successfully!) problem 24 in projectEuler I was doodling around and discoverd the foloowing identity:
$$1+2\times2!+3\times3!+\dots N\times N!=\sum_{k=1}^{k=N} k\times k!=(N+1)!-1$$
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Why limit Euler's Partition function P to $k\leq\sqrt n$ instead of $k\leq n$?
I solved a Project Euler problem (I won't say which one) involving the Partition Function P.
I used equation #11 from the above link:
$$P(n) = \sum_{k=1}^n (-1)^{k+1}\bigg(P\Big(n-{1\over ...
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102 views
What is the relationship between these expression?
Moderator Note: This is a Project Euler question
If ...
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2answers
140 views
using markov chains to solve a project-euler problem?
I never learned what markov chain is, but from googling it seems like if there are finite states and each state has probabilities to jump to other states, I can use markov chain.
What I'm on is ...
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Solutions exist for $\left\lceil \sqrt{n^2 - x^2} \right\rceil - \left\lceil \sqrt{(n-1)^2 - x^2} \right\rceil = 2$ for $n$ big enough
Does there always exist an integer solution $x \in [0,n-1]$ to the equation
$$\left\lceil \sqrt{n^2 - x^2} \right\rceil - \left\lceil \sqrt{(n-1)^2 - x^2} \right\rceil = 2$$
when $n \ge 8$? A computer ...
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1answer
135 views
Searching for pandigital numbers
I was working on the Euler project's problems and the 32nd problem is the following:
We shall say that an n-digit number is pandigital if it makes use of
all the digits 1 to n exactly once; for ...
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5answers
258 views
Every odd composite $=$ prime ${}+ 2x^2$
I was looking through some project-euler questions and I came across one that said
Every odd composite number can be written as the sum of a prime and twice a square...This was proven false.
...
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3answers
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Find the largest prime factor
I just "solved" the third Project Euler problem:
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
With this on Mathematica:
...
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3answers
350 views
10 most significant digits of the sum of a 100 50-digit numbers
This is about Project Euler #13. You are given a 100 50-digit numbers and are asked to calculate the 10 most significant digits of the sum of the numbers.
The solution threads stated that we are only ...
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370 views
How many integer solutions to a diophantine equation
Starting with the equation:
$\frac{1}{a}+\frac{1}{b}=\frac{p}{10^n}$,
I reached the equation:
$10^{n-log(p)} = \frac{ab}{a+b}$.
Now given the positive integer $n$, for what integer values of $p$ ...
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2answers
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Solving $x^2 \cdot y^2 + x^2 + y^2 = c^2$ with $x$, $y$, $c \in \mathbb{Z}^+$
I am working on Project Euler 390.
The question is about triangles, and finding the area of a triangle with sides $\sqrt{a^2+1}, \sqrt{b^2+1}$ and $\sqrt{a^2+b^2}$, with $a, b \in \mathbb{Z}$. I have ...
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2answers
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Solving $\phi (n) < (n-1) \cdot \frac{15499}{94744} $
I am working on challenge 243 from Project Euler (PE 243). The question is:
$$\text{Solve } \phi (n) < (n-1)\cdot \frac{15499}{94744}$$
I can calculate $\phi(n)$ for any $n$, but I think the $n$ ...
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4answers
165 views
Identifying which “plain text” is in English
I need to identify which text is in English from a list of possible list of plain texts generated from a brute force attack on a cipher text. I am thinking of using frequency distributions of English ...
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4answers
351 views
The longest sum of consecutive primes that add to a prime less than 1,000,000
In Project Euler problem $50,$ the goal is to find the longest sum of consecutive primes that add to a prime less than $1,000,000. $
I have an efficient algorithm to generate a set of primes ...
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1answer
132 views
Algorithm for generating an ordered list of pair products
For problem 4 in the euler project part of the assignment is to generate a list of products of 3-digit numbers.
The easy way is to just do a cartesian product (I think it's called), and after that ...
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2answers
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Compute $\sum_0^{n-1}2^i11^{n-i-1}\bmod10^9$ when $n=13^{17}$
Given the following function $f$
$f(1)=1$
$f(n)=11\cdot f(n-1)+2^{n-1}$
I would like to compute $f(13^{17})\mod 10^9$ and ended up using the following :
$f(n)=\sum_{i=0}^{n-1}({11^{n-(i+1)}\cdot ...
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1answer
160 views
Primality and repeated digits
I recently worked on problem 51 through project euler, I solved it essentially through brute-force but afterwards I viewed the forum and there were some more clever solutions.
For those unfamiliar ...
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2answers
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Steps to get Inverse of Pentagonal
I have solved http://projecteuler.net/problem=44 by getting the inverse equation from Wikipedia http://en.wikipedia.org/wiki/Pentagonal_number:
Pentagonal:
$f(n) = \frac{n(3n - 1)}{2}$
Inverse ...
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1answer
221 views
How to find large prime factors without using computer?
What is the largest prime factor of the number 600851475143 ?
This is the third problem of Project Euler.
How to approach this mathematically (without computer programming) ?
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1answer
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How time consuming is this integer optimization problem? [closed]
Minimize $x$ subject to:
$$\tag 1x^2 - Dy^2 =1$$
$$\tag 2 x,y>0$$
$$\tag 3 x,y\in \Bbb Z$$
$$\tag 4 2<D\leqslant 1000$$ (except when D is square)
I am playing with Project Euler and trying ...
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2answers
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sum of even-valued and odd-valued Fibonacci numbers
I was solving the Project Euler problem 2
*By starting with 1 and 2, the first 10 terms of Fibonacci Series will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Find the sum of all the even-valued terms ...
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Euler's Numerical method help
Consider this differential equation,
$dy/dx = x + \sin(y)$
with initial condition $y = 0.5$ when $x = 1.2$:
Write down the recurrence relation for Euler's numerical method
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1answer
269 views
Finding all possible paths from one corner to the other on a grid, without backtracking
Me again "new to maths guy". Please tell me if the substance of my questions are not a good fit for the site.
I'm now onto Question 15 of Project Euler and it seems like there's some mathematical ...
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1answer
125 views
How to find the factors of numbers around 1e7?
I don't have a maths background but I'm solving problems on the awesome Project Euler .net in JavaScript as programming practice.
I don't want to link directly to the question or post it verbatim ...
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Counting couples having least common multiple less than a number
Let f(n) be the number of couples (x,y) with x and y positive integers, $x\leq y$ and the least common multiple of x and y equal to n.
Let g be the summatory function of f, i.e.:
$g(n) = ...
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1answer
194 views
How does $P(n)$ (Partition Function P) work?
I am trying to do Project Euler #78, which is about the different ways of splitting up coins into piles. I realized that this is just the number of integer partitions of the number of coins, ...
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1answer
297 views
Comparing Powers with Different Bases Using Logarithms?
I looked all over to see if a question like this had already been answered, but I couldn't find it. So here goes:
I need a general formula for comparing two (insanely huge) powers. I'm pretty sure ...
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Does “triangle” in English exclude degenerate triangles?
Just for fun read few problems on the projeteuler.net website.
Number 276 found interesting:
Consider the triangles with integer sides a, b and c with a ≤ b ≤ c.
An integer sided triangle ...
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1answer
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Project Euler Problem 371
Project Euler Problem 371 states
Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).
While driving to work Seth plays the following ...
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1answer
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For what kind of numbers would $r_2(n^2) = 420$?
I am trying to find all of the answers to $r_2(n^2) = 420$, where $N < 10^{11}$. It is for finding lattice points on a circle with points $(0,0), (N,0), (0,N)$, and $(N,N)$. I am (pretty) sure that ...
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4answers
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Flaw in expected value solving logic (Project Euler 323)
The problem statement for Project Euler #323 is as follows:
Let $y_0, y_1, y_2, ...$ be a sequence of random unsigned 32 bit integers (i.e. $0 \leq y_i < 2^{32}$, every value equally likely).
...
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2answers
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Combinations/Permutations Count Paths Through Grid
I am curious about a situation in permutations/combinations. This question stems from a challenge site (project euler, problem 15) and research found on this exchange and elsewhere. The question ...
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Project Euler Question 222
Would I be wrong to assume that the solution to this problem:
What is the length of the shortest pipe, of internal radius 50mm, that can fully contain 21 balls of radii 30mm, 31mm, ..., 50mm?
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3answers
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Generating Numeric Palindromes.
I have just started the Euler project, and felt like I didn't get the fourth problem right...I used string conversion to test if my numbers were symmetrical, instead of relying on (the much faster) ...
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Comparing powers without logarithms
Related to this question and this Project Euler problem (Problem 99), I came up with a recursive algorithm for comparing two numbers of the form $x^y$ (with $x>1$ and $y\ge 0$) without explicit use ...
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Bell-like recurrence
Let
$$A(n)=\sum_{k=0}^{n-1}\binom{n}{k}A(k)+n!,\quad A(0)=1$$
$$B(n)=\sum_{k=0}^{n-1}\binom{n}{k}B(k)-n!-n!\sum_{k=1}^{n}\frac{1}{k!},\quad B(0)=-1.$$
I'm interested in computing $S(n)=A(n)+B(n)$ ...
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66 views
What is the max possible value of the sum of power of y of each digits?
I'm trying to solve the 30th euler problem. My code is working, but I'm not sure if it's luck or ingeniousness.
To be the most efficient, I want to reduce at the maximum the numbers to checks.
I ...
