# Square root digital expansion

*Author: Andrei Osipov*

https://projecteuler.net/problem=80

It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all. The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475.

For the first one hundred natural numbers, find the total of the digital sums of the first one hundred decimal digits for all the irrational square roots.

The following algorithm was used for the solution: http://www.afjarvis.staff.shef.ac.uk/maths/jarvisspec02.pdf

Source code: prob080-andreoss.pl

use v6; my constant $limit = 100; sub sqrt-subtraction($n) { my Int $a = $n * 5 ; my Int $b = 5; while $b < 10 * 10 ** $limit { given $a <=> $b { when More | Same { # replace a with a − b, and add 10 to b. $a -= $b; $b += 10; } when Less { # add two zeros to a $a *= 100; # add a zero to b just before the final digit (which will always be ‘5’). $b = ($b - (my $x = $b % 10)) * 10 + $x; } } } $b; } sub MAIN(Bool :$verbose = False) { say [+] do for 1 ... 100 -> $n { next if $n.sqrt.floor ** 2 == $n; my $x = [+] $n.&sqrt-subtraction.comb[^$limit]; say "$n $x" if $verbose; $x; } say "Done in {now - INIT now}" if $verbose; }