Two typical examples are showing the non-solvability of the Diophantine equation r < sup > 2 </ sup > + s < sup > 4 </ sup > = t < sup > 4 </ sup > and proving that any prime p such that p ≡ 1 ( mod 4 ) can be expressed as a sum of two squares.

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers ; an example ( there are others ) is described by the flow chart above and as an example in a later section.

* The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because is the root of.

If the quadratic polynomial is monic then the roots are quadratic integers.

All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

The relation " x is even, y is odd " between a pair ( x, y ) of integers is antisymmetric:

He is reputed to have observed that when the lengths of vibrating strings are expressible as ratios of integers ( e. g. 2 to 3, 3 to 4 ), the tones produced will be harmonious.

In number theory, if P ( n ) is a property of positive integers, and if p ( N ) denotes the number of positive integers n less than N for which P ( n ) holds, and if

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

Similar to the view of Leopold Kronecker that " God made the integers ; all else is the work of man ," musicians drawn to the alphorn and other instruments that sound the natural harmonics, such as the natural horn, consider the notes of the natural harmonic series — particularly the 7th and 11th harmonics — to be God's Notes, the remainder of the chromatic scale enabled by keys, valves, slides and other methods of changing the qualities of the simple open pipe being an artifact of mere mortals.

In standard arrays, each index is restricted to a certain range of consecutive integers ( or consecutive values of some enumerated type ), and the address of an element is computed by a " linear " formula on the indices.

where m and n are nonnegative integers with, is the azimuthal angle in radians, and is the normalized radial distance.

One common version, the two-argument Ackermann – Péter function, is defined as follows for nonnegative integers m and n:

To see that it is sufficient — that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product — one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals.

To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor.

That a / b is rational is a necessary and sufficient condition for the existence of some real number c, and integers m and n, such that

) This result was extended in 1987 to show that the probability that k random integers has greatest common divisor d is d < sup >- k </ sup >/ ζ ( k ).

Likewise, if Q ( x, n ) denotes the number of n-free integers ( e. g. 3-free integers being cube-free integers ) between 1 and x, one can show

Hilbert aimed to show the consistency of mathematical systems from the assumption that the " finitary arithmetic " ( a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial ) was consistent.

In fact, Galois theory can be used to show that they may be expressed as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots.

Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2 < sup > n + a </ sup > + b where a and b are integers, and in particular also for Woodall numbers.

Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2 < sup > n + a </ sup > + b where a and b are integers, and in particular also for Woodall numbers.

Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that a < sup > s </ sup > ≡ a < sup > t </ sup > ( mod n ).

It is easy to show that the map d ↦ Q (√ d ) is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields.

Suppose that the Fermat equation with exponent &# 8467 ; ≥ 3 had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve E. It is an elliptic curve and one can show that its discriminant Δ is equal to 16 ( abc )< sup > 2 &# 8467 ;</ sup > and its conductor N is the radical of abc, i. e. the product of all distinct primes dividing abc.

for some integers m & n. It is easy to show that has to be zero, by analyzing easily observed bounds for this integral:

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.

The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.

( Expressed more technically, in each case the pair ( m, n ) decreases in the lexicographic order on pairs, which is a well-ordering, just like the ordering of single non-negative integers ; this means one cannot go down in the ordering infinitely many times in succession.

An example is the " divides " relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p ( and not with any integer that is not a multiple of p ).

The most commonly used grading systems are the Fontainebleau system which ranges from 1 to 8c +, and the John Sherman V-grade system, beginning at V0 and increasing by integers to a current top grade of proposed V16 ( The Wheel of Life by Dai Koyamada in the Grampians, Australia ; The Game, by Daniel Woods, Boulder Canyon, CO ; Lucid Dreaming, by Paul Robinson.

However, it turns out that the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall.

What we have done here is arranged the integers and the odd integers into a one-to-one correspondence ( or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1. 2 × 10 < sup > 12 </ sup > ( over a trillion ).

An example of an NP-complete problem is the subset sum problem: given a finite set of integers is there a non-empty subset which sums to zero?

An example of a problem which is known to be in NP and in co-NP is integer factorization: given positive integers m and n determine if m has a factor less than n and greater than one.

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6 / π < sup > 2 </ sup > ( see pi ), which is about 61 %.

Suppose n < sub > 1 </ sub >, n < sub > 2 </ sub >, …, n < sub > k </ sub > are positive integers which are pairwise coprime.

Rather than excluding the special case n = 0, it is more useful to include ( which, as mentioned before, is isomorphic to the ring of integers ), for example when discussing the characteristic of a ring.

A special case of a random function is a random permutation, where a realisation can be interpreted as being in the form of a function on the set of integers describing the original location of an item, where the value of the function provides the new ( permuted ) location of the item that was in a given location.

The Gaussian integers are a special case of the quadratic integers.

Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers a and b, the algorithm runs in no more than 5k steps, where k is the number of ( decimal ) digits of b. He also proved a special case of Fermat's last theorem.

Dividing integers in a computer program requires special care.

In mathematics, a unique factorization domain ( UFD ) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements ( or irreducible elements ), analogous to the fundamental theorem of arithmetic for the integers.

Algebraic integers are a special case of integral elements of a ring extension.

In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Prüfer domain ; it turns out that the ring of algebraic integers is slightly more special than this: it is a Bézout domain.

This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n < sub > 1 </ sub >,..., n < sub > c </ sub >, there is a number, R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >), such that if the edges of a complete graph of order R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order n < sub > i </ sub > whose edges are all colour i. The special case above has c

The period lattices are of a very special form, being proportional to the Gaussian integers.

This group of transformations is isomorphic to the projective special linear group PSL ( 2, Z ), which is the quotient of the 2-dimensional special linear group SL ( 2, Z ) over the integers by its center

Most functions indicate that they detected an error by returning a special value, typically NULL for functions that return pointers, and-1 for functions that return integers.

For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ < sub > 2 </ sub > is non-constant, because the special fiber isn't smooth.

The special number field sieve is efficient for integers of the form r < sup > e </ sup > ± s, where r and s are small ( for instance Mersenne numbers ).

are relatively prime positive integers, then ( a − 1 )( b − 1 ) − 1 is the largest number that is not a sum of nonnegative multiples of a and b. This is a special case of the Coin Problem.

An antimagic square is a special kind of heterosquare where the 2n + 2 row, column and diagonal sums are consecutive integers.

In particular, the circle group, the additive group Z < sub > p </ sub > of p-adic integers, compact special unitary groups SU ( n ) and all finite groups have property ( T ).

He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.

Folkman's theorem, the statement that there exist arbitrarily large sets of integers all of whose nonempty sums are monochromatic, may be seen as a special case of Rado's theorem concerning the regularity of the system of equations

They did conjecture that this was the case and showed that it would follow from a special case of the four exponentials conjecture in transcendental number theory, specifically that for any two distinct prime numbers p and q, the only real numbers t for which both p < sup > t </ sup > and q < sup > t </ sup > are rational are the positive integers.