Project Euler is a series of challenging mathematical/computer programming problems. Please see the site and rules before posting.

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Non-Recursive Fundamental Recurrence Formulas

Is there a non-recursive version of the fundamental recurrence formulas for continued fractions? I am trying to compute $A_{1000}$, and it is taking me an extremely long time. By the way, I am ...
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3answers
21k views

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
2k 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. ...
3
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1answer
511 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
307 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
458 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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2answers
321 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
2k 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
160 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 ...
6
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3answers
515 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
548 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
87 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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0answers
115 views

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 ...
4
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2answers
708 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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1answer
441 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
677 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
14k views

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: ...
6
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3answers
2k 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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0answers
756 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 $p$ ...
3
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2answers
319 views

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
232 views

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
274 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
2k 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
323 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
106 views

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
807 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
109 views

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
2k 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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2answers
3k views

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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1answer
151 views

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
780 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 ...
2
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1answer
600 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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0answers
472 views

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
348 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
1k 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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2answers
274 views

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 ...
2
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1answer
888 views

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 ...
2
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1answer
686 views

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 ...
4
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4answers
705 views

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
4k views

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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3answers
776 views

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
3k views

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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1answer
705 views

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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0answers
293 views

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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1answer
95 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 ...
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2answers
240 views

What is wrong with my algorithm (finding if the origin is within a triangle's interior)?

I am working on Project Euler Problem 102 and I thought I had a solution, but it seems I do not. Now, don't give me the solution. I know I'm on the right track. What I want to know is why my method ...
2
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2answers
2k views

Computing Non-zero End Digits of Large Factorials

Any large factorial will have a number of zero behind it, and one could write an expression to compute the number of trailing zeros, but how would one go about computing the non-zero end digits? E.g. ...
2
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1answer
1k views

Project Euler 215 - Solving with Sparse Matrices and Vector Multiplication

I wrote a program to solve Project Euler Problem #215 (see below for description) using memoization, and when I got access to the PE forums, I saw everyone else wrote programs that also used dynamic ...
2
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1answer
410 views

How is this probability found (project euler #121)?

In the problem statement of project Euler problem 121, the following information is given: A pouch contains one black chip and one white chip. In a simple game, one player takes a chip at random and ...
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0answers
719 views

Project Euler Problem 338

I'm stuck on Project Euler problem 338. This is a cross post from StackOverflow where I initially posted, however, it was suggested that I post it here too since the problem mostly relies on math. The ...