Articles of project euler

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.: $g(n) = \sum_{i=1}^{n}f(i)$. I am looking for advice on a formula or an algorithm to calculate g(n) in […]

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) ?

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 $(-1+i)$. For example, Project Euler lists the following conversions: $$ (0+0i)_{10} = (0)_{-1+i} \\ (-5+0i)_{10} = (11001101)_{-1+i} \\ (8+0i)_{10} = […]

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? …involves stacking the balls, from largest to smallest, with each ball resting against the last on alternate sides of […]

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 13195 are 5, 7, 13, and 29. What is the largest prime factor of the number 600851475143 ? As a caveat, in accordance with the […]

Project Euler, Problem #25

Problem #25 from Project Euler asks: What is the first term in the Fibonacci sequence to contain 1000 digits? The brute force way of solving this is by simply telling the computer to generate Fibonacci numbers until it finds the first one that has 1000 digits. I wanted to look for a more mathematical solution […]

Largest prime factor of 600851475143

This question already has an answer here: How to find large prime factors without using computer? 2 answers Find the largest prime factor 4 answers

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 are not included. So I […]

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 approach this problem for a given N?

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: Select[Divisors[600851475143], PrimeQ] It will first give me a list with all the divisors of 600851475143 and then It’s going to select […]