Step by Step solution about How to resolve the algorithm Taxicab numbers step by step in the Julia programming language

The provided Julia code finds and prints taxi cab numbers up to a specified limit. Taxi cab numbers are positive integers that can be expressed as the sum of two cubes in two different ways. The code achieves this efficiently using dictionaries and sorting algorithms.

Detailed Explanation:

1. Function findtaxinumbers:

   • Input: nmax - The upper limit for finding taxi cab numbers.
   • Creates dictionaries cube2n and sum2cubes to optimize calculations.

2. Creating cube2n Dictionary:

   • This dictionary maps each cube value (x^3) to the original number x.

3. Creating sum2cubes Dictionary:

   • This dictionary stores sets of number pairs that sum up to each possible sum of two cubes.

4. Populating sum2cubes and Finding Taxi Cab Numbers:

   • Iterates over all possible pairs of cube values, calculating their sum (c1 + c2).
   • For each sum c, checks if it already exists in sum2cubes.
   • If not, creates a new set. If it does, adds the pair (cube2n[c1], cube2n[c2]) to the existing set.

5. Collecting and Sorting Taxi Cab Numbers:

   • Collects all entries from sum2cubes that have more than one pair.
   • Sorts these taxi cab numbers by their cube sum.

6. Printing Taxi Cab Numbers:

   • Prints the first 25 and last 6 taxi cab numbers up to nmax along with their cube representations.

7. Alternative findtaxinumbers Function:

   • This version uses a different approach to find taxi cab numbers, improving efficiency.
   • Creates cubes and crev dictionaries for storing cubes and their corresponding numbers.
   • Iterates over the cubes and sums of cubes, checking if a given sum can be expressed as the sum of two different cubes. If so, prints the result.

Additional Notes:

• The code uses Dict and Set data structures from the DataStructures package for efficient storage and manipulation.
• It utilizes IterTools.product for iterating over all possible pairs of dictionary entries.
• The @printf macro is used for formatted printing.

using Printf, DataStructures, IterTools

function findtaxinumbers(nmax::Integer)
    cube2n = Dict{Int,Int}(x ^ 3 => x for x in 0:nmax)
    sum2cubes = DefaultDict{Int,Set{NTuple{2,Int}}}(Set{NTuple{2,Int}})
    for ((c1, _), (c2, _)) in product(cube2n, cube2n)
        if c1 ≥ c2
            push!(sum2cubes[c1 + c2], (cube2n[c1], cube2n[c2]))
        end
    end

    taxied = collect((k, v) for (k, v) in sum2cubes if length(v) ≥ 2)
    return sort!(taxied, by = first)
end
taxied = findtaxinumbers(1200)

for (ith, (cube, set)) in zip(1:25, taxied[1:25])
    @printf "%2i: %7i = %s\n" ith cube join(set, ", ")
    # println(ith, ": ", cube, " = ", join(set, ", "))
end
println("...")
for (ith, (cube, set)) in zip(2000:2006, taxied[2000:2006])
    @printf "%-4i: %i = %s\n" ith cube join(set, ", ")
end

# version 2
function findtaxinumbers(nmax::Integer)
    cubes, crev = collect(x ^ 3 for x in 1:nmax), Dict{Int,Int}()
    for (x, x3) in enumerate(cubes)
        crev[x3] = x
    end
    sums = collect(x + y for x in cubes for y in cubes if y < x)
    sort!(sums)

    idx = 0
    for i in 2:(endof(sums) - 1)
        if sums[i-1] != sums[i] && sums[i] == sums[i+1]
            idx += 1
            if 25 < idx < 2000 || idx > 2006 continue end
            n, p = sums[i], NTuple{2,Int}[]
            for x in cubes
                n < 2x && break
                if haskey(crev, n - x)
                    push!(p, (crev[x], crev[n - x]))
                end
            end
            @printf "%4d: %10d" idx n
            for x in p @printf(" = %4d ^ 3 + %4d ^ 3", x...) end
            println()
        end
    end
end

findtaxinumbers(1200)