number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, a balanced prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

with equal-sized prime gaps above and below it, so that it is equal to the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...

of the nearest primes above and below. Or to put it algebraically, given a prime number

p_n

, where is its index in the ordered set of prime numbers, :

p_n = .

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

Examples

The first few balanced primes are 5, 53,

157 Year 157 ( CLVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Civica and Aquillus (or, less frequently, year 910 ''Ab urbe condit ...

173 Year 173 ( CLXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Pompeianus (or, less frequently, year 926 ''Ab urbe ...

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

257 __NOTOC__ Year 257 ( CCLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Gallienus (or, less frequently, year 10 ...

, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 .

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. the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is:The Largest Known CPAP's
Retrieved on 2014-06-13. :

p_n = 1213266377 \times 2^ + 2429,\quad  p_ = p_n-2430,\quad  p_ = 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 = .

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 sequences , , and in the OEIS respectively.

References

{{Prime number classes Classes of prime numbers Unsolved problems in number theory

Examples

Infinitude

Generalization

See also

References