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.

Acid

An acid (from the Latin acidus/acēre meaning sour) is a chemical substance whose aqueous solutions are characterized by a sour taste, the ability to turn blue litmus red, and the ability to react with bases and certain metals (like calcium) to form salts. An aqueous solution of an acid has a pH of less than 7 and is colloquially also referred to as 'acid' (as in 'dissolved in acid'), while the strict definition refers only to the solute. An acid usually contains a hydrogen atom bonded to a chemical structure that is still energetically favorable after loss of H⁺ (a positive hydrogen ion or proton). A lower pH means a higher acidity, and thus a higher concentration of positive hydrogen ions in the solution. Chemicals or substances having the property of an acid are said to be acidic.

There are three common definitions for acids: the Arrhenius definition, the Brønsted-Lowry definition, and the Lewis definition. The Arrhenius definition defines acids as substances which increase the concentration of hydrogen ions (H⁺), or more accurately, hydronium ions (H₃O⁺), when dissolved in water. The Brønsted-Lowry definition is an expansion to include solvents other than water: an acid is a substance which can act as a proton donor. By this definition, any compound which can be deprotonated can be considered an acid. Examples include alcohols and amines which contain O-H or N-H fragments. A Lewis acid is a substance that can accept a pair of electrons to form a covalent bond. Examples of Lewis acids include all metal cations, and electron-deficient molecules such as boron trifluoride and aluminium trichloride.

Waves

Waves are oscillations acompanied by a transfer of energy that travel through space or mass.

Waves may also refer to:

See also