ATYPIC

Comparing two numbers written in index form like 211 and 37 is not difficult, as any calculator would confirm that 211 = 2048 519432525806 would be much more difficult, as both numbers contain over three million digits. Using base_exp.txt, a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which lin..

Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6. n Relatively Prime φ(n) n/φ(n) 2 1 1 2 3 1, 2 2 1.5 4 1, 3 2 2 5 1, 2, 3, 4 4 1.25 6 1, 5 2 3 7 1, 2, 3, 4, 5, 6 6 1.1666… 8 1, 3, 5, 7 4 2 9 1..

In the 5 by 5 matrix above, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to 2427. \begin{pmatrix} \color{red}{131} & 673 & 234 & 103 & 18\\ \color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & 150\\ 630 & 803 & \color{red}{746} & \color{red}{422} & 111\\ 537 & 699 & 497 & \color{red}{121} & 956\\ ..