Wilson prime

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p² divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 2×10¹³. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y)/log(x)).

Several computer searches have been done in the hope of finding new Wilson primes. The Ibercivis distributed computing project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.

Generalizations

Wilson primes of order n

Wilson's theorem can be expressed in general as for every prime . Generalized Wilson primes of order n are the primes p such that divides .

It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n.

Harold Wilson

James Harold Wilson, Baron Wilson of Rievaulx, KG, OBE, PC, FRS, (11 March 1916 – 24 May 1995) was a British Labour Party politician who served as the Prime Minister of the United Kingdom from 1964 to 1970 and 1974 to 1976.

First entering Parliament in 1945, Wilson was immediately appointed the Parliamentary Secretary to the Ministry of Works and rose quickly through the ministerial ranks, becoming the Secretary for Overseas Trade in 1947 and being appointed to the Cabinet just months later as the President of the Board of Trade. Later, in the Labour Shadow Cabinet, he served first as Shadow Chancellor of the Exchequer from 1955 to 1961 and then as the Shadow Foreign Secretary from 1961 to 1963, when he was elected Leader of the Labour Party after the sudden death of Hugh Gaitskell. Wilson narrowly won the 1964 election, going on to win a much increased majority in a snap 1966 election.

Wilson's first period as Prime Minister coincided with a period of low unemployment and relative economic prosperity, though also of significant problems with Britain's external balance of payments. In 1969 Wilson sent British troops to Northern Ireland. After losing the 1970 general election to Edward Heath, he spent four years as Leader of the Opposition before the February 1974 general election resulted in a hung parliament. After Heath's talks with the Liberals broke down, Wilson returned to power as leader of a minority government until there was a second general election in the autumn, which resulted in a narrow Labour victory. A period of economic crisis was now beginning to hit most Western countries, and in 1976 Wilson suddenly announced his resignation as Prime Minister.

Prime minister

A prime minister is the most senior minister of cabinet in the executive branch of government, often in a parliamentary or semi-presidential system. In many systems, the prime minister selects and may dismiss other members of the cabinet, and allocates posts to members within the government. In most systems, the prime minister is the presiding member and chairman of the cabinet. In a minority of systems, notably in semi-presidential systems of government, a prime minister is the official who is appointed to manage the civil service and execute the directives of the head of state.

In parliamentary systems fashioned after the Westminster system, the prime minister is the presiding and actual head of government and head of the executive branch. In such systems, the head of state or the head of state's official representative (i.e. the monarch, president, or governor-general) usually holds a largely ceremonial position, although often with reserve powers.

The prime minister is often, but not always, a member of parliament and is expected with other ministers to ensure the passage of bills through the legislature. In some monarchies the monarch may also exercise executive powers (known as the royal prerogative) that are constitutionally vested in the crown and may be exercised without the approval of parliament.

Wilson

Wilson may refer to:

Name

Leaders

Places

Australia

Canada

Poland

United Kingdom

United States of America

Prime number

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. are all valid factorizations of 3.

The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits.