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

The provided Julia code, focuses on counting Pythagorean triplets within specified perimeter limits, specifically exploring both all possible and "primitive" triplets. Here's a detailed breakdown:

1. primitiven{T<:Integer}(m::T) Function:

   • This function generates a vector of primitive Pythagorean triples with a given perimeter m.
   • A primitive triplet is one where the greatest common divisor (GCD) of the triplet's three sides is 1.
   • The function starts by checking if m is greater than 1; if not, it returns an empty vector.
   • It then checks if m is equal to 2; if so, it returns a vector containing only the number 1.
   • If m is not prime, it generates a vector of all the primitive triplets with a perimeter of m. To do this, it creates a vector of boolean values, rp, where rp[i] is true if i is a primitive Pythagorean triple.
   • It then iterates through all the factors of m and sets rp[p:p:m-1] to false for each factor p. This removes all the non-primitive triplets from the vector.
   • Finally, it returns a vector containing all the primitive Pythagorean triples with a perimeter of m.

2. pythagoreantripcount{T<:Integer}(plim::T) Function:

   • This function counts the number of all Pythagorean triplets and primitive Pythagorean triplets within a specified perimeter limit plim.
   • It initializes two counters, primcnt and fullcnt, to 0.
   • It then iterates through all possible values of m from 2 to plim.
   • For each value of m, it calculates the corresponding value of p as 2m^2.
   • It then iterates through all the primitive Pythagorean triples with a perimeter of m using the primitiven function.
   • For each primitive Pythagorean triple, it calculates the corresponding value of q as p + 2m*n.
   • It checks if q is less than or equal to plim; if not, it breaks out of the loop.
   • It increments primcnt by 1 and increments fullcnt by the number of times q can divide plim.
   • Finally, it returns a tuple containing the values of primcnt and fullcnt.

3. Main Body:

   • The main body of the code prints a table showing the number of all Pythagorean triplets and primitive Pythagorean triplets within specified perimeter limits.
   • It iterates through the values of om from 1 to 10.
   • For each value of om, it calculates plim as 10^om and calls the pythagoreantripcount function to count the number of all Pythagorean triplets and primitive Pythagorean triplets within the specified perimeter limit.
   • It then prints the values of om, fcnt, and pcnt in a table.

function primitiven{T<:Integer}(m::T)
    1 < m || return T[]
    m != 2 || return T[1]
    !isprime(m) || return T[2:2:m-1]
    rp = trues(m-1)
    if isodd(m)
        rp[1:2:m-1] = false
    end
    for p in keys(factor(m))
        rp[p:p:m-1] = false
    end
    T[1:m-1][rp]
end

function pythagoreantripcount{T<:Integer}(plim::T)
    primcnt = 0
    fullcnt = 0
    11 < plim || return (primcnt, fullcnt)
    for m in 2:plim
        p = 2m^2
        p+2m <= plim || break
        for n in primitiven(m)
            q = p + 2m*n
            q <= plim || break
            primcnt += 1
            fullcnt += div(plim, q)
        end
    end
    return (primcnt, fullcnt)
end

println("Counting Pythagorian Triplets within perimeter limits:")
println("    Limit          All   Primitive")
for om in 1:10
    (pcnt, fcnt) = pythagoreantripcount(10^om)
    println(@sprintf "    10^%02d  %11d   %9d" om fcnt pcnt)
end