Step by Step solution about How to resolve the algorithm AKS test for primes step by step in the Go programming language

The provided Go code is related to mathematics and specifically deals with finding prime numbers using an AKS (Agrawal-Kayal-Saxena) primality test, which is known for its high efficiency in determining prime numbers.

Here's a breakdown of the code:

1. bc Function:

   • This function computes the coefficients of a polynomial that is related to the AKS primality test.
   • It takes an integer p as input and returns a slice of integers c representing the coefficients of the polynomial.
   • The coefficients are calculated using a loop and binomial coefficients.

2. pp Function:

   • This function converts the coefficients of a polynomial (represented as a slice of integers) into a string representation of the polynomial.
   • It handles the case when the polynomial has only one term or multiple terms, using appropriate formatting to represent the sum of terms.

3. aks Function:

   • This function implements the AKS primality test.
   • It takes an integer p as input and returns a boolean true if p is prime; otherwise, it returns false.
   • It does so by modifying the polynomial coefficients and checking if the resulting coefficients are all divisible by p.

4. main Function:

   • The main function is the entry point of the program.
   • It performs two tasks:
     • First, it prints the coefficients of polynomials for various values of p (up to 7) to visually demonstrate the polynomial.
     • Second, it prints prime numbers less than 50 by iterating through integers and applying the AKS test.

In summary, this code demonstrates the AKS primality test in Go. It implements the bc function to compute polynomial coefficients, the pp function to format the polynomial, and the aks function to perform the AKS test for primality. It then uses these functions to print the coefficients of polynomials for specific values of p and find prime numbers less than 50.

package main

import "fmt"

func bc(p int) []int64 {
    c := make([]int64, p+1)
    r := int64(1)
    for i, half := 0, p/2; i <= half; i++ {
        c[i] = r
        c[p-i] = r
        r = r * int64(p-i) / int64(i+1)
    }
    for i := p - 1; i >= 0; i -= 2 {
        c[i] = -c[i]
    }
    return c
}

func main() {
    for p := 0; p <= 7; p++ {
        fmt.Printf("%d:  %s\n", p, pp(bc(p)))
    }
    for p := 2; p < 50; p++ {
        if aks(p) {
            fmt.Print(p, " ")
        }
    }
    fmt.Println()
}

var e = []rune("²³⁴⁵⁶⁷")

func pp(c []int64) (s string) {
    if len(c) == 1 {
        return fmt.Sprint(c[0])
    }
    p := len(c) - 1
    if c[p] != 1 {
        s = fmt.Sprint(c[p])
    }
    for i := p; i > 0; i-- {
        s += "x"
        if i != 1 {
            s += string(e[i-2])
        }
        if d := c[i-1]; d < 0 {
            s += fmt.Sprintf(" - %d", -d)
        } else {
            s += fmt.Sprintf(" + %d", d)
        }
    }
    return
}

func aks(p int) bool {
    c := bc(p)
    c[p]--
    c[0]++
    for _, d := range c {
        if d%int64(p) != 0 {
            return false
        }
    }
    return true
}