# Tagged Questions

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

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### Project Euler's, Problem #565

Project Euler's, Problem #565 states: Let $\sigma(n)$ be the sum of the divisors of $n$. E.g. the divisors of $4$ are $1, 2$ and $4$, so $\sigma(4)=7$. The numbers $n$ not exceeding $20$ ...
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### How many sets correspond to connected graphs

I'm trying to solve this project euler problem. I don't want to get too much help, since that would defeat the purpose, but I'm hitting a wall, so I'm asking a related problem here, from which I'll ...
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### Possible mis-interpretation in Project Euler #21

Here is the problem statement for Problem 21 of Project Euler. Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$). If $d(a) = b$ ...
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### The largest product of two n-digit numbers which is palindrome

Project Euler: 4 is stated as follows: Largest palindrome product A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 ...
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### Game of Nim: Losing Positions [closed]

If you have heard of the game Nim, this is a version of the game. However, in this version, the players can only remove the amount of stones from the pile which is coprime to the current pile size. ...
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### Project Euler 106: Necessary and sufficient conditions

Problem Statement Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are ...
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### Help Project Euler Problem 269

I am stuck on prob 269 Project Euler. I've just tried brute force method to attempt this problem the example provided by PE For example, $P_{5703}(x)$ = $5x^3 + 7x^2 + 3$. We can see that: P$_n(0)$...
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### Proof that nonconstant polynomial cannot have the same value at all integer points

I am reading the solution to the Project Euler problem 28 here, specifically the one under 'Deriving a non-iterative formula'. That solution first deduces the degree of the polynomial, and then ...
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### (Soft Question) Is it bad to use Sage built in functions instead of creating my own?

I've been doing Project-Euler just as a way to increase my competency in computer science. I'm currently a Pure and Applied Math major who recently adopted computer science as a minor in order to ...
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### How to simplify a sum of complex divisors?

This question arises from Project Euler 153. That problem asks for the sum of all complex divisors of all natural numbers up to a maximum, where a complex divisor is a complex number of the form a + ...
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### square of digits - why does it always contain 1 or 89 [closed]

I attempted project euler problem 92, while I passed it, my solution works, but had just...awful performance. So I would like to try again tomorrow. In the meantime understanding why the iteration ...
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### Where is my formula false??

I wrote a formula that returned how many numbers in a given row of pascals triangle are divisible by a given prime. This formula was created to answer https://projecteuler.net/problem=148. I was ...
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### Number of 1s in after converting number to base -1+i

Regarding to Base conversion: How to convert between Decimal and a Complex base? Let $s(a,b)$ is a number of $1$ after converting complex number $a+bi$ to base $-1+i$. It's easy to implement that ...
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### Base conversion: How to convert between Decimal and a Complex base?

My motivation for this question is exploring beyond the ideas in Project Euler Problem 508. In that problem, it is helpful to know how to convert between a decimal number and a number in base $(-1+i)$....
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An addition chain is an ordered tuple of numbers, starting with $1$, such that each number after $1$ can be expressed as the sum of two smaller numbers in the chain. An example of an addition chain ...
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### How to find $a_n$ series from Dirichlet generating function

I am solving problems from Project Euler. Solutions for some of the problems is $n^{\rm th}$ term of a series. I know Dirichlet generating functions. How to find $n^{\rm th}$ term of a series from ...
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### Cubes differences and primality

In an exercise (Project Euler 131, not to mention it), we are looking for perfect cubes of the form $n^3 + n^2 p$, where p is prime. I finally got the answer by trial and error but I don't understand ...
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### Finding prime factors by taking the square root

I'm trying to solve the third Project Euler problem and I'd like a little help understanding a mathematical concept underlying my tentative solution. The question reads: The prime factors of ...
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### Find the sum of all the multiples of 3 or 5 below 1000 (Break down)

I know that this has been posted before but I can't grasp how it actually works. I'm a 16 year old in the 10th grade and am interested in algorithms. I've looked in multiple places on the web but am ...
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### How to compute a slowly converging series to 10 decimals places of accuracy?

I'm looking at a Project Euler problem, where a harmonic series is modified such that it excludes terms where a digit appears three times consecutively in the denominator. So this series would exclude ...
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### Largest prime factor of a number

In Project Euler problem 3, where we have to find the largest prime factor of a number, one of the solution i came across is ...
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### Are there any mathematics “problem websites” similar to Project Euler?

Are there any mathematics websites similar to Projet Euler, a website which hosts math-heavy programming questions, many of which can be solved with a pen and paper? I've become almost addicted to ...
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### Computing question: A quadratic which gives primes [closed]

This is about Project Euler Problem 27. The question is: Considering quadratics of the form $n^2 + an + b$, where $\lvert a \rvert < 1000$ and $\lvert b \rvert < 1000$ Find the product ...
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### Find the sum of all n, 0 < n < 64,000,000 such that σ2(n) is a perfect square.

I have been working on Project Euler problem 211 for quite some time, and I am stuck. I'm not looking for an answer, I'm simply looking for some guidance. I've written and tested the following code, ...
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### Sum of $x^x$ final 10 digits

warning/spoiler alert this problem occurs in the euler project. I want to find the last ten digits of the following sum: $$S = 1^1 + 2^2 + 3^3 + 4^4 + \cdots + 1000^{1000}$$ Finding this ...
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### Number of paths through a grid

I am attempting to solve Project Euler prob. 204 by hand, and I decided that this method should help a lot. I have the following grid: The numbers indicate the total number of paths from a node at ...
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### What rules can I apply for divisible numbers?

I'm trying to solve Project Euler problem 12. I've figured out a fast algorithm for the triangle numbers, but to see if they are devisable is just a pain... ...
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### How does this algorithm find the largest prime factor?

This question on math.stackexchange details an algorithm that can be used to find the largest prime factor of a number. I used it to solve Project Euler #3. Here's a short description of the algorithm:...
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### Digit Factorial Sum? [duplicate]

This question is founded in Project Euler #34. I originally solved the problem years ago but now I'm moving all the problems over to a new language. As I revisit this problem, I already know the ...
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### How to find all pairs $(a, b)$ s.t. $(a^2+b^2)/\gcd(a,b) \leq n$ for constant $n$?

Any help is appreciated, this is for my work on http://projecteuler.net/problem=153. Also posted here
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### Why does $n^2 \equiv 10 \pmod{30}$ imply $n \equiv 0 \pmod{10}$?

It seems that $n^2 \equiv 10 \pmod{30} \iff n \equiv 0 \pmod{10}$. I found this by calculating $\{n \in \mathbb N_0 \mid n < 30 \land n^2 \equiv 10 \pmod{30}\} = \{10, 20\}$, and noting that 10 ...
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### Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
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### Project Euler #453 confusion

So I decided to give a shot on the #453 project euler problem but there is something that confuses me with the numbers given. I decided to start by calculating the possible arrangements of 4 vertices ...
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### How to find positive integers where the multiplicative modular inverse is equal to itself for mod n?

This a question sparked from Project Euler Question. I really devoted so much time to search an efficient solution however no output. What are some possibles theorems or formulas that are useful in ...
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### could not able to understand Project Euler 18. “Maximum path sum I”

According to question, By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23. ...
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### Project Euler - 34 / Find a mathematical approach for upper bound

145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145. Find the sum of all numbers which are equal to the sum of the factorial of their digits. Note: as 1! = 1 and 2! = 2 are not sums they ...
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### repunit prime factors

So I am working on this problem... which states: 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 ...
### An Euler problem: How many of these numbers are of the form $a^b$?
How much numbers can be written in the form $a^b$, where $a$ and $b$ are integers that are between $2$ and $100$? How can I start this problem? Any hints please? Thanks!