How to resolve the algorithm Pythagorean triples step by step in the Tcl programming language

Published on 12 May 2024 09:40 PM

How to resolve the algorithm Pythagorean triples step by step in the Tcl programming language

Table of Contents

Problem Statement

A Pythagorean triple is defined as three positive integers

( a , b , c )

{\displaystyle (a,b,c)}

where

a < b < c

{\displaystyle a<b<c}

, and

a

2

b

2

=

c

2

.

{\displaystyle a^{2}+b^{2}=c^{2}.}

They are called primitive triples if

a , b , c

{\displaystyle a,b,c}

are co-prime, that is, if their pairwise greatest common divisors

g c d

( a , b )

g c d

( a , c )

g c d

( b , c )

1

{\displaystyle {\rm {gcd}}(a,b)={\rm {gcd}}(a,c)={\rm {gcd}}(b,c)=1}

. Because of their relationship through the Pythagorean theorem, a, b, and c are co-prime if a and b are co-prime (

g c d

( a , b )

1

{\displaystyle {\rm {gcd}}(a,b)=1}

).   Each triple forms the length of the sides of a right triangle, whose perimeter is

P

a + b + c

{\displaystyle P=a+b+c}

.

The task is to determine how many Pythagorean triples there are with a perimeter no larger than 100 and the number of these that are primitive.

Deal with large values.   Can your program handle a maximum perimeter of 1,000,000?   What about 10,000,000?   100,000,000? Note: the extra credit is not for you to demonstrate how fast your language is compared to others;   you need a proper algorithm to solve them in a timely manner.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Pythagorean triples step by step in the Tcl programming language

Source code in the tcl programming language

proc countPythagoreanTriples {limit} {
    lappend q 3 4 5
    set idx [set count [set prim 0]]
    while {$idx < [llength $q]} {
    set a [lindex $q $idx]
    set b [lindex $q [incr idx]]
    set c [lindex $q [incr idx]]
    incr idx
    if {$a + $b + $c <= $limit} {
        incr prim
        for {set i 1} {$i*$a+$i*$b+$i*$c <= $limit} {incr i} {
        incr count
        }
        lappend q \
        [expr {$a + 2*($c-$b)}] [expr {2*($a+$c) - $b}] [expr {2*($a-$b) + 3*$c}] \
        [expr {$a + 2*($b+$c)}] [expr {2*($a+$c) + $b}] [expr {2*($a+$b) + 3*$c}] \
        [expr {2*($b+$c) - $a}] [expr {2*($c-$a) + $b}] [expr {2*($b-$a) + 3*$c}]
    }
    }
    return [list $count $prim]
}
for {set i 10} {$i <= 10000000} {set i [expr {$i*10}]} {
    lassign [countPythagoreanTriples $i] count primitive
    puts "perimeter limit $i => $count triples, $primitive primitive"
}


  

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