Step by Step solution about How to resolve the algorithm Earliest difference between prime gaps step by step in the Go programming language

Overview

This program finds the earliest pair of prime gaps with a difference greater than a specified limit, starting from the beginning of the prime sequence.

Detailed Explanation

• Imports: The program imports the fmt package for input and output and the rcu package for prime generation and number formatting.

• Constants and Variables:

  • limit: A constant representing the maximum limit for the prime search space (1e9).
  • gapStarts: A map that stores the starting primes for each prime gap.
  • primes: A slice containing the first limit * 5 prime numbers.

• Prime Generation: The rcu.Primes function is used to generate the first limit * 5 prime numbers and store them in the primes slice. This provides a sufficient range of primes for the search.

• Prime Gap Analysis:

  • The program iterates through the prime numbers, calculating the gap between each consecutive pair using primes[i] - primes[i-1].
  • If a gap has not been seen before, its starting prime is added to the gapStarts map.

• Main Loop: The main loop continues indefinitely, searching for prime gaps with a difference greater than the current limit (pm).

  • It starts with a gap of 2 and searches for the earliest gap that starts at a prime larger than the current starting prime.
  • When such a gap is found, the program calculates the difference between its starting prime and the starting prime of the previous gap.
  • If the difference exceeds the limit, the program prints the details of the gap and updates the limit to 10 times its previous value.
  • If the difference is less than or equal to the limit, the program moves on to the next gap.

Output: The program outputs the details of the earliest prime gaps with a difference greater than the specified limit, including the gap size, starting primes, and difference. It continues searching for gaps with increasingly larger differences until reaching the limit of 1e9.

package main

import (
    "fmt"
    "rcu"
)

func main() {
    limit := int(1e9)
    gapStarts := make(map[int]int)
    primes := rcu.Primes(limit * 5)
    for i := 1; i < len(primes); i++ {
        gap := primes[i] - primes[i-1]
        if _, ok := gapStarts[gap]; !ok {
            gapStarts[gap] = primes[i-1]
        }
    }
    pm := 10
    gap1 := 2
    for {
        for _, ok := gapStarts[gap1]; !ok; {
            gap1 += 2
        }
        start1 := gapStarts[gap1]
        gap2 := gap1 + 2
        if _, ok := gapStarts[gap2]; !ok {
            gap1 = gap2 + 2
            continue
        }
        start2 := gapStarts[gap2]
        diff := start2 - start1
        if diff < 0 {
            diff = -diff
        }
        if diff > pm {
            cpm := rcu.Commatize(pm)
            cst1 := rcu.Commatize(start1)
            cst2 := rcu.Commatize(start2)
            cd := rcu.Commatize(diff)
            fmt.Printf("Earliest difference > %s between adjacent prime gap starting primes:\n", cpm)
            fmt.Printf("Gap %d starts at %s, gap %d starts at %s, difference is %s.\n\n", gap1, cst1, gap2, cst2, cd)
            if pm == limit {
                break
            }
            pm *= 10
        } else {
            gap1 = gap2
        }
    }
}