Appendix K — Goldbach’s Conjecture

K.1 Statement of the Conjecture

Goldbach’s Conjecture is one of the oldest and most famous unsolved problems in number theory. It was first proposed by the German mathematician Christian Goldbach in a letter to Leonhard Euler in 1742.

K.1.1 Strong Goldbach Conjecture

Definition K.1 (Strong Goldbach Conjecture) Every even integer greater than 2 can be expressed as the sum of two prime numbers.

Formally, for every even integer \(n > 2\), there exist prime numbers \(p\) and \(q\) such that:

\[ n = p + q \]

K.1.2 Weak Goldbach Conjecture

Definition K.2 (Weak Goldbach Conjecture) Every odd integer greater than 5 can be expressed as the sum of three prime numbers.

Formally, for every odd integer \(n > 5\), there exist prime numbers \(p\), \(q\), and \(r\) such that:

\[ n = p + q + r \]

K.2 Historical Context

• 1742: Christian Goldbach proposed the conjecture in a letter to Leonhard Euler
• 1937: Ivan Vinogradov proved that every sufficiently large odd integer can be expressed as the sum of three primes, making significant progress toward the weak conjecture
• 2013: Harald Helfgott completed the proof of the weak Goldbach conjecture
• Present: The strong Goldbach conjecture remains unproven, though it has been verified computationally for all even numbers up to extremely large values

K.3 Examples

Example K.1 (Small even numbers) Here are some examples of even numbers expressed as the sum of two primes:

• \(4 = 2 + 2\)
• \(6 = 3 + 3\)
• \(8 = 3 + 5\)
• \(10 = 3 + 7 = 5 + 5\)
• \(12 = 5 + 7\)
• \(14 = 3 + 11 = 7 + 7\)
• \(16 = 3 + 13 = 5 + 11\)
• \(18 = 5 + 13 = 7 + 11\)
• \(20 = 3 + 17 = 7 + 13\)

Note that for most even numbers, there are multiple ways to express them as the sum of two primes.

K.4 Computational Verification

# Note: This code prioritizes clarity and educational value over performance.
# For production use with large numbers, consider optimizations such as:
# - Sieve of Eratosthenes for generating primes
# - Pre-allocating data structures instead of using rbind()

# Function to check if a number is prime
is_prime <- function(n) {
  if (n < 2) return(FALSE)
  if (n == 2) return(TRUE)
  if (n %% 2 == 0) return(FALSE)
  if (n == 3) return(TRUE)
  
  # Check odd divisors from 3 to sqrt(n)
  i <- 3
  while (i * i <= n) {
    if (n %% i == 0) return(FALSE)
    i <- i + 2
  }
  return(TRUE)
}

# Function to find Goldbach pairs for an even number
find_goldbach_pairs <- function(n) {
  if (n <= 2 || n %% 2 != 0) {
    return(NULL)
  }
  
  pairs <- list()
  for (p in 2:(n / 2)) {
    q <- n - p
    if (is_prime(p) && is_prime(q)) {
      pairs[[length(pairs) + 1]] <- c(p, q)
    }
  }
  return(pairs)
}

# Test the conjecture for even numbers from 4 to 100
verify_goldbach <- function(max_n = 100) {
  results <- data.frame(
    n = integer(),
    num_pairs = integer(),
    first_pair = character(),
    stringsAsFactors = FALSE
  )
  
  for (n in seq(4, max_n, by = 2)) {
    pairs <- find_goldbach_pairs(n)
    if (length(pairs) > 0) {
      first_pair_str <- paste(pairs[[1]], collapse = " + ")
      results <- rbind(results, data.frame(
        n = n,
        num_pairs = length(pairs),
        first_pair = first_pair_str,
        stringsAsFactors = FALSE
      ))
    }
  }
  
  return(results)
}

# Verify for even numbers up to 100
goldbach_results <- verify_goldbach(100)
print(goldbach_results)
#>      n num_pairs first_pair
#> 1    4         1      2 + 2
#> 2    6         1      3 + 3
#> 3    8         1      3 + 5
#> 4   10         2      3 + 7
#> 5   12         1      5 + 7
#> 6   14         2     3 + 11
#> 7   16         2     3 + 13
#> 8   18         2     5 + 13
#> 9   20         2     3 + 17
#> 10  22         3     3 + 19
#> 11  24         3     5 + 19
#> 12  26         3     3 + 23
#> 13  28         2     5 + 23
#> 14  30         3     7 + 23
#> 15  32         2     3 + 29
#> 16  34         4     3 + 31
#> 17  36         4     5 + 31
#> 18  38         2     7 + 31
#> 19  40         3     3 + 37
#> 20  42         4     5 + 37
#> 21  44         3     3 + 41
#> 22  46         4     3 + 43
#> 23  48         5     5 + 43
#> 24  50         4     3 + 47
#> 25  52         3     5 + 47
#> 26  54         5     7 + 47
#> 27  56         3     3 + 53
#> 28  58         4     5 + 53
#> 29  60         6     7 + 53
#> 30  62         3     3 + 59
#> 31  64         5     3 + 61
#> 32  66         6     5 + 61
#> 33  68         2     7 + 61
#> 34  70         5     3 + 67
#> 35  72         6     5 + 67
#> 36  74         5     3 + 71
#> 37  76         5     3 + 73
#> 38  78         7     5 + 73
#> 39  80         4     7 + 73
#> 40  82         5     3 + 79
#> 41  84         8     5 + 79
#> 42  86         5     3 + 83
#> 43  88         4     5 + 83
#> 44  90         9     7 + 83
#> 45  92         4     3 + 89
#> 46  94         5     5 + 89
#> 47  96         7     7 + 89
#> 48  98         3    19 + 79
#> 49 100         6     3 + 97

NoteComputational Evidence

The strong Goldbach conjecture has been verified computationally for all even integers up to at least \(4 \times 10^{18}\) (as of 2020). While this provides strong empirical evidence, it does not constitute a mathematical proof.

K.5 Current Status

• Weak Goldbach Conjecture: Proven (Harald Helfgott, 2013)
• Strong Goldbach Conjecture: Unproven (remains an open problem)

The strong Goldbach conjecture is considered one of the most important unsolved problems in mathematics. Despite centuries of effort by mathematicians and extensive computational verification, a general proof remains elusive.

K.6 Relevance to Applied Mathematics

While Goldbach’s conjecture itself is a problem in pure mathematics (number theory), studying such problems develops important skills:

• Understanding the relationship between conjectures and proofs
• Distinguishing between empirical evidence and mathematical proof
• Working with number-theoretic concepts
• Developing computational verification methods

These skills are valuable in applied mathematics and statistics, where we often work with theoretical results that must be verified empirically.

K.7 References

Additional resources on Goldbach’s conjecture:

• (hardy2008introduction?)
• https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
• https://mathworld.wolfram.com/GoldbachConjecture.html