After a sum of the first billion primes post (originally from Programming Praxis), I decided to finally write a segmented version of the Sieve of Eratosthenes.
After a sum of the first billion primes post (originally from Programming Praxis), I decided to finally write a segmented version of the Sieve of Eratosthenes.
Programming Praxis’ new challenge(s) are to write three different list algorithms three times, each with a different runtime complexity. From their first post last week we have list intersection and union and from a newer post yesterday we have the difference of two lists. For each of those, we want to be able to write an algorithm that runs in O(n2) time, one that runs in O(n log n), and finally one that runs in O(n). It turns out that it’s more of an exercise in data structures than anything (although they’re all still technically ’list’ algorithms), but it’s still interesting to see how you can achieve the same goal in different ways that may be far more efficient.
A Pythagorean triplet is a set of three natural numbers, a b c, for which,
a2 + b2 = c2
For example, 32 + 42 = 9 + 16 = 25 = 52.
There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc. – PROJECT EULER #9
Find the greatest product of five consecutive digits in the 1000-digit number. – PROJECT EULER #8
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number? – PROJECT EULER #7
The sum of the squares of the first ten natural numbers is,
12 + 22 + … + 102 = 385
The square of the sum of the first ten natural numbers is,
(1 + 2 + … + 10)2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum. – PROJECT EULER #6
Yesterday had another programming puzzle by Programming Praxis. This time, we are looking for a very special sort of number, a Taxicab number. According to Wikipedia:
In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways. – Wikipedia: Taxicab Number
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20? – PROJECT EULER #5
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 x 99.
Find the largest palindrome made from the product of two 3-digit numbers. – PROJECT EULER #4
Each new term in the Fibonacci sequence is generated by adding the previous two terms.
By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms. – Project Euler #2