Balanced prime

In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number $p_{n}$ , where n is its index in the ordered set of prime numbers,

p_{n}={{p_{{n-1}}+p_{{n+1}}} \over 2}.

Examples

The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 (sequence A006562 in the OEIS).

For example, 53 is the sixteenth prime. The fifteenth and seventeenth primes, 47 and 59, add up to 106, half of which is 53, thus 53 is a balanced prime.

When 1 was considered a prime number, 2 would have correspondingly been considered the first balanced prime since

2={1+3 \over 2}.

Infinitude

It is conjectured that there are infinitely many balanced primes.

Three consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. As of 2014 the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is:^[1]

p_{n}=1213266377\times 2^{{35000}}+2429,\quad p_{{n-1}}=p_{n}-2430,\quad p_{{n+1}}=p_{n}+2430.

The value of n (its rank in the sequence of all primes) is not known.

Generalization

The balanced primes may be generalized to the balanced primes of order n. A balanced prime of order n is a prime number that is equal to the arithmetic mean of the nearest n primes above and below. Algebraically, given a prime number $p_{k}$ , where k is its index in the ordered set of prime numbers,

p_{k}={\sum _{{i=1}}^{n}({p_{{k-i}}+p_{{k+i}})} \over 2n}.

Thus, an ordinary balanced prime is a balanced prime of order 1.The sequences of balanced primes of orders 2, 3, and 4 are given as (sequence A082077 in the OEIS), (sequence A082078 in the OEIS), and (sequence A082079 in the OEIS) respectively.

References

↑ The Largest Known CPAP's. Retrieved on 2014-06-13.

Prime number classes
By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)
By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Almost prime Semiprime Interprime
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229
List of prime numbers

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Balanced prime

Examples

Infinitude

Generalization

See also

References