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http://projecteuler.net/problem=66 Diophantine equationProblem 66 Consider quadratic Diophantine equations of the form: x2 – Dy2 = 1 For example, when D=13, the minimal solution in x is 6492 – 131802 = 1. It can be assumed that there are no solutions in positive integers when D is square. By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following: 32 – 222 = 1 22 – 312 = ..

http://projecteuler.net/problem=99 Largest exponentialProblem 99 Comparing two numbers written in index form like 211 and 37 is not difficult, as any calculator would confirm that 211 = 2048 37 = 2187. However, confirming that 632382518061 519432525806 would be much more difficult, as both numbers contain over three million digits. Using base_exp.txt (right click and 'Save Link/Target As...'), a..

Maximum path sum II Problem 67Published on Saturday, 10th April 2004, 02:00 am; Solved by 46063 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. 3 7 4 2 4 6 8 5 9 3 That is, 3 + 7 + 4 + 9 = 23. Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text fi..

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http://projecteuler.net/problem=65 Problem 65 12 March 2004 The square root of 2 can be written as an infinite continued fraction. 2 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + ... The infinite continued fraction can be written, 2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, 23 = [4;(1,3,1,8)].It turns out that the sequence of partial values of continued fractions for square roots p..

http://projecteuler.net/problem=64 Problem 64 27 February 2004 All square roots are periodic when written as continued fractions and can be written in the form: N = a0 + 1 a1 + 1 a2 + 1 a3 + ... For example, let us consider 23: 23 = 4 + 23 — 4 = 4 + 1 = 4 + 1 1 23—4 1 + 23 – 3 7 If we continue we would get the following expansion: 23 = 4 + 1 1 + 1 3 + 1 1 + 1 8 + ... The process can be summarise..

http://projecteuler.net/problem=63 Problem 63 13 February 2004 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? Answer: 49 * Racket #lang racket (define (digits n) (add1 (floor (/ (log n) (log 10))))) (define sum 0) (define d 1) (for* ([n (in-range 1 10)] [d (in..

http://projecteuler.net/problem=62 Problem 62 30 January 2004 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 also cube.Find the smallest cube for which exactly five permutations of its digits are cube. Answer: 127035954683 * Racket #lang..