Problem #63 concerns perfect powers and the number of digits in the perfect power. The question reads: Project Euler Problem 63: Powerful digit counts The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power. How many n-digit positive integers exist which are also an nth power? My solution is …
Tag Archives: Project Euler
Project Euler Problem #62
Problem #62 concerns perfect cubes whose digits are rearrangements of each other. The question reads: Project Euler Problem 62: Cubic permutations The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are …
Project Euler Problem #61
Problem #61 involves the figurate numbers. The question reads: Project Euler Problem 61: Cyclical figurate numbers Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae: Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, … Square P4,n=n2 1, 4, 9, 16, 25, …
Project Euler Problem #60
Problem #60 is one of the many Project Euler problems that involves manipulating digits in prime numbers. The question reads: Project Euler Problem 60: Prime pair sets The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For …
Project Euler Problem #59
Problem #59 concerns a type of cipher in cryptography known as XOR encryption. The question reads: Project Euler Problem 59: XOR decryption Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and …
Project Euler Problem #58
Problem #58 is another example of a Project Euler problem where the process of Engineer’s Induction can be very useful. The question reads: Project Euler Problem 58: Spiral primes Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed. 37 36 35 34 33 32 31 38 17 16 15 …
Project Euler Problem #57
Problem #57 is the first of several Project Euler problems to involve infinite continued fractions for irrational numbers, and it is the first of several problems that can b killed with Pell Equations. The problem reads: Project Euler Problem 57: Square root convergents It is possible to show that the square root of two can …
Project Euler Problem #56
Problem #56 concerns the sum of the digits of various perfect powers. The question reads: Project Euler Problem 56: Powerful digit sum A googol (10100) is a massive number: one followed by one-hundred zeros; 100100 is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is …
Project Euler Problem #55
Problem #55 involves an obscure type of number called a Lychrel number. The question reads: Project Euler Problem 55: Lychrel numbers If we take 47, reverse and add, 47 + 74 = 121, which is palindromic. Not all numbers produce palindromes so quickly. For example, 349 + 943 = 1292, 1292 + 2921 = 4213 …
Project Euler Problem #53
Problem #53 is one of the many problems in Project Euler that involves binomial coefficients. The question reads: Project Euler Problem 53: Combinatoric selections There are exactly ten ways of selecting three from five, 12345: 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345 In combinatorics, we use the notation, (5 choose 3)=10. …
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