Balanced Primes

A balanced prime is a prime number that is equal to the arithmetic mean of the nearest primes above and below it. In other words, a prime number is balanced if:

where and are the previous and next prime numbers respectively. In other words, for a prime to be balanced, it must satisfy , and the gaps between consecutive primes must be .

The first few balanced primes are 5 (balanced between 3 and 7), 53 (balanced between 47 and 59), 157 (balanced between 151 and 163), 173 (balanced between 167 and 179).

Computational Complexity

The process of finding balanced primes involves checking the primality of numbers, which can be efficiently done using the Sieve of Eratosthenes. This algorithm has a time complexity of , where is the upper limit of numbers being checked. Additionally, the space complexity is , as it requires storing an array of boolean values to represent the primality of each number up to .

code.js
1function isBalancedPrime(n) {
2  const primes = sieveOfEratosthenes(n);
3  const primeIndex = primes.indexOf(n);
4  const prevPrime = primes[primeIndex - 1];
5  const nextPrime = primes[primeIndex + 1];
6  return n === (prevPrime + nextPrime) / 2;
7}
8
9function sieveOfEratosthenes(n) {
10  const isPrime = Array(n + 1).fill(true);
11  isPrime[0] = isPrime[1] = false;
12  for (let i = 2; i * i <= n; i++) {
13    if (isPrime[i]) {
14      for (let j = i * i; j <= n; j += i) {
15        isPrime[j] = false;
16      }
17    }
18  }
19  return isPrime.map((isPrime, index) => isPrime ? index : null).filter(Boolean);
20}

Applications

Balanced primes have applications in:

• Cryptography
• Number theory research
• Prime gap analysis
• Mathematical pattern recognition

Interesting Facts

1. Balanced primes are relatively rare compared to regular primes
2. The distribution of balanced primes becomes sparser as numbers get larger
3. There are infinitely many balanced primes (proven)
4. The largest known balanced prime has over 1000 digits