# Solution of;
# Project Euler Problem 585: Nested square roots
# https://projecteuler.net/problem=585
# 
# Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a 
# nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ 
# are not allowed to be perfect squares, so the number below the outer square 
# root is irrational. Still it can be shown that for some combinations of $x$, 
# $y$ and $z$ the given term can be simplified into a sum and/or difference of 
# simple square roots of integers, actually denesting the square roots in the 
# initial expression. Here are some examples of this denesting:$\small 
# \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1$$\small 
# \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}$$\small 
# \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}-\sqrt{2}=3+\sqrt{6}+\sqrt{3}-\sqrt{2}$$\small 
# \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}-\sqrt{2}$As you 
# can see the integers used in the denested expression may also be perfect 
# squares resulting in further simplification. Let F($n$) be the number of 
# different terms $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$, that can be denested 
# into the sum and/or difference of a finite number of square roots, given the 
# additional condition that $0<x \le n$. That is,$\small \displaystyle 
# \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}$with $k$, $x$, $y$, 
# $z$ and all $a_i$ being positive integers, all $s_i =\pm 1$ and $x\le n$. 
# Furthermore $y$ and $z$ are not allowed to be perfect squares. Nested roots 
# with the same value are not considered different, for example $\small 
# \sqrt{7+\sqrt{3}+\sqrt{27}}$, $\small \sqrt{7+\sqrt{12}+\sqrt{12}}$ and 
# $\small \sqrt{7+\sqrt{27}+\sqrt{3}}$, that can all three be denested into 
# $\small 2+\sqrt{3}$, would only be counted once. You are given that 
# F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and 
# F(5000)=11134074. Find F(5000000).
# 
# by lcsm29 http://github.com/lcsm29/project-euler
import timed


def dummy(n):
    pass


if __name__ == '__main__':
    n = 1000
    i = 10000
    prob_id = 585
    timed.caller(dummy, n, i, prob_id)

# Solution of;
# Project Euler Problem 585: Nested square roots
# https://projecteuler.net/problem=585
# 
# Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a 
# nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ 
# are not allowed to be perfect squares, so the number below the outer square 
# root is irrational. Still it can be shown that for some combinations of $x$, 
# $y$ and $z$ the given term can be simplified into a sum and/or difference of 
# simple square roots of integers, actually denesting the square roots in the 
# initial expression. Here are some examples of this denesting:$\small 
# \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1$$\small 
# \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}$$\small 
# \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}-\sqrt{2}=3+\sqrt{6}+\sqrt{3}-\sqrt{2}$$\small 
# \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}-\sqrt{2}$As you 
# can see the integers used in the denested expression may also be perfect 
# squares resulting in further simplification. Let F($n$) be the number of 
# different terms $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$, that can be denested 
# into the sum and/or difference of a finite number of square roots, given the 
# additional condition that $0<x \le n$. That is,$\small \displaystyle 
# \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}$with $k$, $x$, $y$, 
# $z$ and all $a_i$ being positive integers, all $s_i =\pm 1$ and $x\le n$. 
# Furthermore $y$ and $z$ are not allowed to be perfect squares. Nested roots 
# with the same value are not considered different, for example $\small 
# \sqrt{7+\sqrt{3}+\sqrt{27}}$, $\small \sqrt{7+\sqrt{12}+\sqrt{12}}$ and 
# $\small \sqrt{7+\sqrt{27}+\sqrt{3}}$, that can all three be denested into 
# $\small 2+\sqrt{3}$, would only be counted once. You are given that 
# F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and 
# F(5000)=11134074. Find F(5000000).
# 
# by lcsm29 http://github.com/lcsm29/project-euler
import timed


def dummy(n):
    pass


if __name__ == '__main__':
    n = 1000
    i = 10000
    prob_id = 585
    timed.caller(dummy, n, i, prob_id)

# Solution of; # Project Euler Problem 585: Nested square roots # https://projecteuler.net/problem=585 # # Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a # nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ # are not allowed to be perfect squares, so the number below the outer square # root is irrational. Still it can be shown that for some combinations of $x$, # $y$ and $z$ the given term can be simplified into a sum and/or difference of # simple square roots of integers, actually denesting the square roots in the # initial expression. Here are some examples of this denesting:$\small # \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1$$\small # \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}$$\small # \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}-\sqrt{2}=3+\sqrt{6}+\sqrt{3}-\sqrt{2}$$\small # \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}-\sqrt{2}$As you # can see the integers used in the denested expression may also be perfect # squares resulting in further simplification. Let F($n$) be the number of # different terms $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$, that can be denested # into the sum and/or difference of a finite number of square roots, given the # additional condition that $0<x \le n$. That is,$\small \displaystyle # \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}$with $k$, $x$, $y$, # $z$ and all $a_i$ being positive integers, all $s_i =\pm 1$ and $x\le n$. # Furthermore $y$ and $z$ are not allowed to be perfect squares. Nested roots # with the same value are not considered different, for example $\small # \sqrt{7+\sqrt{3}+\sqrt{27}}$, $\small \sqrt{7+\sqrt{12}+\sqrt{12}}$ and # $\small \sqrt{7+\sqrt{27}+\sqrt{3}}$, that can all three be denested into # $\small 2+\sqrt{3}$, would only be counted once. You are given that # F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and # F(5000)=11134074. Find F(5000000). # # by lcsm29 http://github.com/lcsm29/project-euler