Euler's totient function

In number theory, Euler's totient function (or Euler's phi function), denoted as φ(n) or ϕ(n), is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. (These integers are sometimes referred to as totatives of n.) Thus, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) = 1.

Euler's totient function is a multiplicative function, meaning that if two numbers m and n are coprime, then φ(mn) = φ(m) φ(n).

For example, let n = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤ k ≤ 9, that is 1, 2, 4, 5, 7 and 8 are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.

Euler's phi function is important mainly because it gives the order of the multiplicative group of integers modulo n (the group of units of the ring ℤ/nℤ). It also plays a key role in the definition of the RSA encryption system.

Number (magazine)

Number is the leading Japanese sports magazine published on every Thursday by Bungeishunju. The official name is Sports Graphic Number. Though the circulation once hit 47 million, it is now around 10 million.

The first issue, released in April 1980, drew attention by the piece Enatsu's 21 balls (江夏の21球, Enatsu no 21 kyu). Yet the magazine failed to return a profit for the next 10 years. Today however, the magazine is one of the most profitable publications of Bungeishunju. The success of the magazine also led other rival publishers to launch sports magazines, though they tend to be less successful.

References

Personal Communications Services area codes

Area codes 500, 533, 544, 566 and 577 are non-geographic area codes reserved for personal communications services. The 500 code was authorized by the United States Federal Communications Commission, and introduced into the North American Numbering Plan in 1995.

In 1995 AT&T introduced a "follow-me" service under the brand name of AT&T True Connections using the 500 area code. It was designed to replace the AT&T EasyReach 700 service. Other local exchange carriers and interexchange carriers introduced similar competitive services.

AT&T True Connections service was not well received. Companies, hotels, and others with PBX equipment continued to block the dialing of 500 because it was a caller-paid number. Phone-sex services also began using the 500 prefix and forwarding the calls to various foreign countries.

The 500 prefix is still registered to a variety of phone companies, and is being used for non-geographic services including dial-up modem access.

Non-geographic telephone numbers in the United Kingdom

A non-geographic number is a type of telephone number that is not linked to any specific locality. Such numbers are an alternative to the traditional 'landline' numbers that are assigned geographically using a system of location-specific area codes. Non-geographic numbers are used for various reasons, from providing flexible routeing of incoming phone calls to generating revenue for paid-for services.

Functionality

Non-geographic numbers were introduced to offer services that were historically unavailable on standard landline phone numbers, particularly in terms of call routeing and special charging arrangements. Advertised benefits of non-geographic numbers include: