Problem 7 asks: What is the 10,001st prime number? My approach uses a straightforward primality test with trial division up to the square root of each candidate number. Here's the complete Python solution: python import  math def is_prime (n):      if  n < 2 :          return False           for  i in range ( 2 , int (math.sqrt(n)) + 1 ):          if  n %  i == 0 :              return False           return True prime_list = [] for  i in range ( 0 , 999999 ):     

Found the difference between the sum of squares and the square of sums for the first 100 natural numbers. The problem: Calculate (1² + 2² + ... + 100²) vs (1 + 2 + ... + 100)² python sum_of_squares = [] sum_of_numbers = [] for  i in range ( 0 , 101 ):     sum_of_squares.append(i * i)     sum_of_numbers.append(i) ss = sum (sum_of_squares) sn = sum (sum_of_numbers) sn2 =  sn *  sn print ( abs (ss -  sn2)) Result: 25,164,150 The mathematical elegance here is that while bo

Problem: Starting in the top-left corner of a 20×20 grid, and only being able to move right and down, how many routes are there to the bottom-right corner? At first glance, this seems like a classic dynamic programming or pathfinding problem. But I took a different approach - let the computer explore small cases, then use pattern recognition to crack the larger problem. from itertools import permutations import math from math import factorial dictionary_x = {} list_x = [0, 1

Problem: Find the sum of all numbers less than one million which are palindromic in both base 10 and base 2 (binary). Note that palindromic numbers can't include leading zeros in either base. My approach was straightforward - check every number up to 1,000,000: python double_base_palindromes = [] for  i in range ( 0 , 1000000 ):     y = list ( str (i))     y.reverse()     x = '' .join(y)      if int (x) == int (i):         b = bin (i)         b =  b[ 2 :]         c =

Tackled another Project Euler challenge - this time finding all numbers that can be written as the sum of the fifth powers of their digits. The problem excludes 1 (as per the problem statement), and asks: which numbers equal the sum of their digits raised to the fifth power? For example: 4150 = 4⁵ + 1⁵ + 5⁵ + 0⁵ = 1024 + 1 + 3125 + 0 = 4150 My Python solution iterates through numbers up to 9,999,999, converts each to individual digits, calculates the sum of fifth powers, and