Balanced prime

Balanced prime

A balanced prime is a prime number that 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}.

The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 OEIS|id=A006562.

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}.

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 2005 the largest known CPAP-3 has 7535 digits found by David Broadhurst and François Morain: [http://hjem.get2net.dk/jka/math/cpap.htm#k3] :p_n = 197418203 imes 2^{25000} - 1, p_{n-1} = p_n-6090, p_{n+1} = p_n+6090.The value of "n" is not known.

ee also

When a prime is greater than the arithmetic mean of its two neighboring primes, it is called a strong prime. When it is less, it is called a weak prime.


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Prime Minister of Pakistan — Infobox Political post post = Prime Minister body = the Islamic Republic of Pakistan insignia = FlagofPrimeMinister.jpg insigniasize = 150px insigniacaption = Flag of the Prime Minister incumbent = Syed Yousaf Raza Gillani incumbentsince = 25… …   Wikipedia

  • Prime Minister of the United States — There is no Prime Minister of the United States, but the term Prime Minister has sometimes been applied, either as a pejorative term, a bon mot , or through ignorance, to an official within the government of the United States.During and after the …   Wikipedia

  • Prime Minister of Croatia — President of the Government of Croatia Predsjednik Vlade Republike Hrvatske Coat of Arms of Croatia …   Wikipedia

  • Strong prime — In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory. Definition in cryptography In cryptography, a prime number p is strong if the… …   Wikipedia

  • Mersenne prime — Named after Marin Mersenne Publication year 1536[1] Author of publication Regius, H. Number of known terms 47 Conjectured number of terms Infinite …   Wikipedia

  • Cuban prime — A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is: and the first few cuban primes from this equation are (sequence A002407 in OEIS): 7 …   Wikipedia

  • 1000 (number) — List of numbers Integers ← 1k 2k 3k 4k 5k 6k 7k 8k 9k → Cardinal 1000 one thousand …   Wikipedia

  • 6000 (number) — 6000 (six thousand) is the natural number following 5999 and preceding 6001. Selected numbers in the range 6001 6999 * 6025 Rhythm guitarist of the Dead Kennedys from June 1978 to March 1979. Full name is Carlos Cadona. * 6028 centered heptagonal …   Wikipedia

  • 4000 (number) — 4000 (four thousand) is the natural number following 3999 and preceding 4001. It is a decagonal number. Selected numbers in the range 4001 4999 * 4005 triangular number * 4007 safe prime * 4010 magic constant of n times; n normal magic square and …   Wikipedia

  • 5000 (number) — 5000 (five thousand) is the natural number following 4999 and preceding 5001. Five thousand is the largest isogrammic number in the English language. Selected numbers in the range 5001 5999 * 5003 Sophie Germain prime * 5020 amicable number with… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”