Balanced prime number

In number theory is a balanced prime (from the English balanced prime ) is a prime number , which precisely between the previous prime number and the following prime number lies. The following applies to the arithmetic mean : ${\displaystyle p_{n}\in \mathbb {P} }$ ${\displaystyle p_{n-1}}$ ${\displaystyle p_{n+1}}$

${\displaystyle p_{n}={\frac {p_{n-1}+p_{n+1}}{2}}}$

Examples

• The 16th is prime . Your prime neighbors are and . The arithmetic mean of these two neighbors is . Thus, is a balanced prime. ${\displaystyle p_{16}=53}$ ${\displaystyle p_{15}=47}$ ${\displaystyle p_{17}=59}$ ${\displaystyle {\frac {p_{15}+p_{17}}{2}}={\frac {47+59}{2}}=53=p_{16}}$ ${\displaystyle p_{16}=53}$
• The smallest balanced prime numbers are the following:

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393, ... (sequence A006562 in OEIS )

• The largest known balanced prime is the following prime:

${\displaystyle p_{n}=1213266377\cdot 2^{35000}+2429}$

She has jobs and was discovered by David Broadhurst in 2014 with the PrimeForm and Primo programs . Your prime neighbors are and . But it is not yet known (see prime number theorem ), so one does not yet know what the prime number is. ${\displaystyle 10546}$ ${\displaystyle p_{n-1}=p_{n}-2430}$ ${\displaystyle p_{n+1}=p_{n}+2430}$ ${\displaystyle \pi (p_{n})}$ ${\displaystyle p_{n}}$

Designations

If you compare a prime number with the arithmetic mean of its prime neighbors and , you get the following types: ${\displaystyle p_{n}}$ ${\displaystyle {\frac {p_{n-1}+p_{n+1}}{2}}}$ ${\displaystyle p_{n-1}}$ ${\displaystyle p_{n+1}}$

• Is is called a strong prime number . ${\displaystyle p_{n}>{\frac {p_{n-1}+p_{n+1}}{2}}}$ ${\displaystyle p_{n}}$

It is closer to the next prime than to the previous prime . ${\displaystyle p_{n+1}}$ ${\displaystyle p_{n-1}}$

• Is , so one calls balanced prime (from the English balanced prime ). ${\displaystyle p_{n}={\frac {p_{n-1}+p_{n+1}}{2}}}$ ${\displaystyle p_{n}}$

It lies exactly between the next prime number and the previous prime number . ${\displaystyle p_{n+1}}$ ${\displaystyle p_{n-1}}$

• If , then one calls a weak prime number (from the English weak prime , not to be confused with the term " weak prime " (from the English weakly prime )). ${\displaystyle p_{n}<{\frac {p_{n-1}+p_{n+1}}{2}}}$ ${\displaystyle p_{n}}$

It is closer to the previous prime than to the next prime . ${\displaystyle p_{n-1}}$ ${\displaystyle p_{n+1}}$

properties

• In an arithmetic prime number sequence with three prime numbers, a balanced prime number is (by definition) the second prime number.

Unsolved problems

• It is believed that there are an infinite number of balanced prime numbers.

Generalizations

A balanced prime number of order k is a prime number which is equal to the arithmetic mean of the neighboring prime numbers below and above. In other words: ${\displaystyle p_{n}}$ ${\displaystyle k}$

${\displaystyle p_{n}={\frac {\sum _{i=1}^{k}(p_{n-i}+p_{n+i})}{2k}}}$

Examples

• The 2931st prime number is . Their smaller prime neighbors are and , the larger prime neighbors are and . The arithmetic mean of these eight neighboring prime numbers is ${\displaystyle p_{2931}=26713}$ ${\displaystyle p_{2927}=26693,p_{2928}=26699,p_{2929}=26701}$ ${\displaystyle p_{2930}=26711}$ ${\displaystyle p_{2932}=26717,p_{2933}=26723,p_{2934}=26729}$ ${\displaystyle p_{2935}=26731}$

${\displaystyle {\frac {p_{2927}+p_{2928}+p_{2929}+p_{2930}+p_{2932}+p_{2933}+p_{2934}+p_{2935}}{8}}={\frac {26693+26699+26701+26711+26717+26723+26729+26731}{8}}={\frac {213704}{8}}=26713=p_{2931}}$

Thus a balanced prime is of order . ${\displaystyle p_{2931}=26713}$ ${\displaystyle 4}$

• The smallest balanced prime numbers of the order are the prime numbers: ${\displaystyle 0}$

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, 233, 239, 241, 251, 257, 263, 269, 271, ... (sequence A000040 in OEIS )

• The smallest balanced prime numbers of the order are the following: ${\displaystyle 2}$

79, 281, 349, 439, 643, 677, 787, 1171, 1733, 1811, 2141, 2347, 2389, 2767, 2791, 3323, 3329, 3529, 3929, 4157, 4349, 4751, 4799, 4919, 4951, 5003, 5189, 5323, 5347, 5521, 5857, 5861, 6287, 6337, 6473, 6967, 6997, 7507, 7933, 8233, 8377, 8429, 9377, 9623, 9629, 10093, 10333, ... (sequence A082077 in OEIS )

• The smallest balanced prime numbers of the order are the following: ${\displaystyle 3}$

17, 53, 157, 173, 193, 229, 349, 439, 607, 659, 701, 709, 977, 1153, 1187, 1301, 1619, 2281, 2287, 2293, 2671, 2819, 2843, 3067, 3313, 3539, 3673, 3727, 3833, 4013, 4051, 4517, 4951, 5101, 5897, 6079, 6203, 6211, 6323, 6679, 6869, 7321, 7589, 7643, 7907, ... (sequence A082078 in OEIS )

• The smallest balanced prime numbers of the order are the following: ${\displaystyle 4}$

491, 757, 1787, 3571, 6337, 6451, 6991, 7741, 7907, 8821, 10141, 10267, 10657, 12911, 15299, 16189, 18223, 18701, 19801, 19843, 19853, 19937, 21961, 22543, 22739, 22807, 23893, 23909, 24767, 25169, 25391, 26591, 26641, 26693, 26713, ... (sequence A082079 in OEIS )

• The smallest balanced prime numbers of order with are the following: ${\displaystyle n}$ ${\displaystyle n=0,1,2,\ldots }$

2, 5, 79, 17, 491, 53, 71, 29, 37, 983, 5503, 173, 157, 353, 5297, 263, 179, 383, 137, 2939, 2083, 751, 353, 5501, 1523, 149, 4561, 1259, 397, 787, 8803, 8803, 607, 227, 3671, 17443, 57097, 3607, 23671, 12539, 1217, 11087, 1087, 21407, 19759, 953, ... (sequence A082080 in OEIS )

Example:

In the list above, the number is in the 10th position . Thus (the 166th prime) is the smallest balanced prime of the order . Indeed it is . ${\displaystyle 983}$ ${\displaystyle p_{166}=983}$ ${\displaystyle 9}$ ${\displaystyle p_{166}={\frac {\sum _{i=1}^{9}(p_{166-i}+p_{166+i})}{2\cdot 9}}={\frac {17694}{18}}=983}$

properties

• Every balanced prime is (by definition) a balanced prime of the order . ${\displaystyle 1}$

Individual evidence

1. ↑ Jens Kruse Andersen: The Largest Known CPAP's. Accessed July 6, 2018 .  

Web links

• balanced prime . In: PlanetMath . (English)
• Chris K. Caldwell: balanced prime. The Prime Glossary, accessed July 6, 2018 . 
• Giovanni Resta: balanced primes. Numbers Aplenty, accessed July 6, 2018 .