The number theory basics (list) is an index to other articles in this wiki, which together form a body of knowledge called number theory. It is organized with definitions and simple results first, and more substantial results afterwards, so that every article depends on earlier results, but not later ones.
It is written in MathHelp's colloquial language so the narrative flows nicely, but lest you fear losing mathematical rigor, the colloquial language is defined before it is used. Example: MathHelp uses the word number to denote residue class. Along with this use of the word number comes all the usual things we do with numbers, such as finding square roots, reciprocals, and the like, all of which have special meaning stemming from the special meaning of number. An example of this would be "the reciprocal of 2 is −2 (mod 5)", which is true because 2 times −2 is equivalent to 1 (mod 5).
DefinitionsEdit
- number: residue class modulo some (usually prime) number that is clear from context.
- equals: of a number, equals (and the equals sign, =) means equivalent.
- square: quadratic residue; a number that's the square of some other number.
- square root of x (mod p): a number, a, such that a^{2} = x (mod p).
- reciprocal of x (mod p): a number, a, such that ax = 1 (mod p).
- a divides b, a and b not both zero: (written a|b) an integer, k exists such that ak=b.
- GCD(a,b): an integer that divides a and b and is divided by all other integers that also divide a and b.
AxiomsEdit
- well ordering principle of integers: Any non-empty set of nonnegative integers has a smallest element.
Basic principlesEdit
- These include simple theorems, lemmas, and identities.
- pigeon hole principle: if kn+1 pigeons are put in n holes then at least one hole will contain at least k+1 pigeons.
- division algorithm: for integer a and positive integer b there exist unique integers q,r such that a=bq+r and 0≤r<b.
- Bézout's identity: for nonzero integers a,b, there exist integers x,y such that ax+by=GCD(a,b)
- Euclidean algorithm: a method of finding GCD(a,b); the extended Euclidean algorithm also finds the Bézout coefficients x and y such that ax+by=GCD(a,b), equivalent to finding the reciprocal of a (mod b).