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Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers E_n = 1+product_(i=1)^(n)p_i (1) = 1+p_n#, (2) ...

A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if p is a prime and p|ab, then p|a or p|b (where | means divides). A corollary is that ...

The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a ...

For any two integers a and b, suppose d|ab. Then if d is relatively prime to a, then d divides b. This results appeared in Euclid's Elements, Book VII, Proposition 30. This ...

An array of "trees" of unit height located at integer-coordinate points in a point lattice. When viewed from a corner along the line y=x in normal perspective, a quadrant of ...

The sequence of numbers obtained by letting a_1=2, and defining a_n=lpf(1+product_(k=1)^(n-1)a_k) where lpf(n) is the least prime factor. The first few terms are 2, 3, 7, 43, ...

1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line ...

In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 ...

A definition assigns properties to some sort of mathematical object. For example, Euclid's Elements starts with a number of definitions, such as "a line is a breadthless ...

A Euclidean number is a number which can be obtained by repeatedly solving the quadratic equation. Euclidean numbers, together with the rational numbers, can be constructed ...