Step by Step solution about How to resolve the algorithm Multi-base primes step by step in the Julia programming language

The provided Julia code defines a function, maxprimebases, that determines the maximum number of prime-valued bases for base-10 numeric strings of a given length and also finds the corresponding base-10 numeric strings.

Here's a breakdown of the code:

1. Function Signature:

   function maxprimebases(ndig, maxbase)

   The maxprimebases function takes two arguments:

   • ndig: The length of the numeric strings being considered.
   • maxbase: The maximum base to use for the prime evaluations.

2. Initialization:

   maxprimebases = [Int[]]
   nwithbases = [0]

   The maxprimebases array will store the bases that yield prime values for each numeric string. The nwithbases array will store the corresponding base-10 numeric strings.

3. Loop Over Primes:

   for p in div(maxprime + 1, 10):maxprime

   The loop iterates over prime numbers in the range [10^(ndig) - 1, 10^ndig].

4. Extract Digits:

   dig = digits(p)

   The digits of the current prime number p are extracted.

5. Find Prime Bases:

   bases = [b for b in 2:maxbase if (isprime(evalpoly(b, dig)) && all(i -> i < b, dig))]

   This comprehension loop iterates over bases from 2 to maxbase and checks two conditions:

   • isprime(evalpoly(b, dig)): Evaluates the prime number p in base b and checks if the result is prime.
   • all(i -> i < b, dig): Checks if all the digits of p are less than b.

6. Update Results:

   if length(bases) > length(first(maxprimebases))
      maxprimebases = [bases]
      nwithbases = [p]
   elseif length(bases) == length(first(maxprimebases))
      push!(maxprimebases, bases)
      push!(nwithbases, p)
   end

   If the number of prime bases found for the current prime number p is greater than or equal to the current maximum, the maxprimebases and nwithbases arrays are updated accordingly.

7. Print Results:

   alen, vlen = length(first(maxprimebases)), length(maxprimebases)
   println("\nThe maximum number of prime valued bases for base 10 numeric strings of length ",
      ndig, " is $alen. The base 10 value list of ", vlen > 1 ? "these" : "this", " is:")
   for i in eachindex(maxprimebases)
      println(nwithbases[i], " => ", maxprimebases[i])
   end

   After the loop completes, it prints the maximum number of prime-valued bases and the corresponding base-10 numeric strings.

8. Timing Measurement:

   @time for n in 1:6
      maxprimebases(n, 36)
   end

   This code snippet times the execution of the maxprimebases function for n from 1 to 6 with a maximum base of 36.

9. Function Signature (Updated):

   function maxprimebases(ndig, maxbase)

   This modified version of the maxprimebases function takes the same two arguments as before.

10. Updated Implementation: The implementation is similar to the previous version but uses a product-based approach to generate all possible numeric strings of length ndig. It also checks for hopelessness early on to avoid unnecessary computation.

11. Loop Over Numeric Strings:

for tup in Iterators.product([0:maxbase-1 for _ in 1:ndig - 1]..., 1:maxbase-1)

This loop iterates over all numeric strings of length `ndig` using a product of ranges.

12. **Find Prime Bases**:
 ```julia
foundbases = Int[]
for b in maximum(dig)+1:maxbase
    if isprime(evalpoly(b, dig))
        push!(foundbases, b)
    end
    maxbase - b + length(foundbases) < length(maxprimebases) && break # shortcut if hopeless
end

Similar to the previous version, this loop checks for prime bases. However, it uses a maximum-digit-plus-one-based range to improve efficiency.

13. Update Results: The update process is largely the same as in the previous version.

14. Print Results: The code prints the maximum number of prime-valued bases and the corresponding base-10 numeric strings. It also indicates whether the numeric string is represented in base 32 (used for digits greater than 9).

15. Timing Measurement:

for n in 1:5 maxprimebases(n, 36) maxprimebases(n, 62) end

This code snippet measures the execution time of the updated `maxprimebases` function for `n` from 1 to 5 with maximum bases of 36 and 62. 
<div id="sourcecode"/>

## Source code in the julia programming language

```julia  
using Primes

function maxprimebases(ndig, maxbase)
 maxprimebases = [Int[]]
 nwithbases = [0]
 maxprime = 10^(ndig) - 1
 for p in div(maxprime + 1, 10):maxprime
     dig = digits(p)
     bases = [b for b in 2:maxbase if (isprime(evalpoly(b, dig)) && all(i -> i < b, dig))]
     if length(bases) > length(first(maxprimebases))
         maxprimebases = [bases]
         nwithbases = [p]
     elseif length(bases) == length(first(maxprimebases))
         push!(maxprimebases, bases)
         push!(nwithbases, p)
     end
 end
 alen, vlen = length(first(maxprimebases)), length(maxprimebases)
 println("\nThe maximum number of prime valued bases for base 10 numeric strings of length ",
     ndig, " is $alen. The base 10 value list of ", vlen > 1 ? "these" : "this", " is:")
 for i in eachindex(maxprimebases)
     println(nwithbases[i], " => ", maxprimebases[i])
 end
end

@time for n in 1:6
 maxprimebases(n, 36)
end


using Primes

function maxprimebases(ndig, maxbase)
 maxprimebases = [Int[]]
 nwithbases = ["0"]
 for tup in Iterators.product([0:maxbase-1 for _ in 1:ndig - 1]..., 1:maxbase-1)
     dig = collect(tup)
     foundbases = Int[]
     for b in maximum(dig)+1:maxbase
         if isprime(evalpoly(b, dig))
             push!(foundbases, b)
         end
         maxbase - b + length(foundbases) < length(maxprimebases) && break # shortcut if hopeless
     end
     if length(foundbases) > length(first(maxprimebases))
         maxprimebases = [foundbases]
         nwithbases = [prod(string.(reverse(dig), base = any(x -> x > 9, dig) ? 32 : 10))]
     elseif length(foundbases) == length(first(maxprimebases))
         push!(maxprimebases, foundbases)
         push!(nwithbases, prod(string.(reverse(dig), base = any(x -> x > 9, dig) ? 32 : 10)))
     end
 end
 alen, vlen = length(first(maxprimebases)), length(maxprimebases)
 println("\nThe maximum number of prime valued bases for base up to $maxbase numeric strings of length ",
     ndig, " is $alen. The value list of ", vlen > 1 ? "these" : "this", " is:")
 for i in eachindex(maxprimebases)
     println(nwithbases[i], maxprimebases[i][1] > 10 ? "(base 32)" : "", " => ", maxprimebases[i])
 end
end

for n in 1:5
 maxprimebases(n, 36)
 maxprimebases(n, 62)
end