* The gcd is a multiplicative function in the following sense: if a < sub > 1 </ sub > and a < sub > 2 </ sub > are relatively prime, then gcd ( a < sub > 1 </ sub >· a < sub > 2 </ sub >, b ) = gcd ( a < sub > 1 </ sub >, b )· gcd ( a < sub > 2 </ sub >, b ).

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains.

has a solution for x if and only if b is divisible by the greatest common divisor d of a and n ( denoted by gcd ( a, n )).

This is multiplicative if the set C has the property that if a and b are in C, gcd ( a, b )= 1, than ab is also in C. This is the case if C is the set of squares, cubes, or higher powers, or if C is the set of square-free numbers.

An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.

In number theory, Euler's totient or phi function, φ ( n ) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ ( n ) is the number of integers k in the range 1 ≤ k ≤ n for which gcd ( n, k ) = 1.

This means that if gcd ( m, n )

However, if gcd ( v, n ) is neither 1 nor n, then the-addition will not produce a meaningful point on the curve, which shows that our elliptic curve is not a group ( mod n ), but, more importantly for now, gcd ( v, n ) is a non-trivial factor of n.

This calculation is always legal and if the gcd of the-coordinate with is unequal to 1 or, so when simplifying fails, then a non-trivial divisor of is found.

: if gcd ( n, k )

* m and n are coprime ( also called relatively prime ) if gcd ( m, n ) = 1 ( meaning they have no common prime factor ).

It also follows from defining a and b as coprime if gcd ( a, b )= 1, so that gcd ( 1, b )= 1 for any b >= 1.

We also see how the results may be presented, although if n, the number of state variables, is large any tabulation will become cumbersome.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

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

The program tries every string, starting with the shortest, until it finds a string with complexity at least n ( if there is one ), then returns that string ( or goes into an infinite loop if there is no such string ).

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

E. g., if n

E. g., if n =

E. g., if n

* completely additive if a ( mn ) = a ( m ) + a ( n ) for all natural numbers m and n ;

* completely multiplicative if a ( mn ) = a ( m ) a ( n ) for all natural numbers m and n ;

Two whole numbers m and n are called coprime if their greatest common divisor is 1 ; i. e., if there is no prime number that divides both of them.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power one temporarily labels the term X with an index i ( running from 1 to n ), then each subset of k indices gives after expansion a contribution X < sup > k </ sup >, and the coefficient of that monomial in the result will be the number of such subsets.

) The exterior derivative of a k-form in R < sup > 3 </ sup > is defined as the ( k + 1 )- form from above ( and in R < sup > n </ sup > if, e. g., < math >

A real number a is said to be computable if it can be approximated by some computable function in the following manner: given any integer, the function produces an integer k such that:

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

As a consequence of the first point, if a and b are coprime, then so are any powers a < sup > k </ sup > and b < sup > l </ sup >.

It is known that there exists a unique Delaunay triangulation for P, if P is a set of points in general position ; that is, there exists no k-flat containing k + 2 points nor a k-sphere containing k + 3 points, for 1 ≤ k ≤ d − 1 ( e. g., for a set of points in < big > ℝ </ big >< sup > 3 </ sup >; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere ).

It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

In 1966, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if, where a < sub > i </ sub > ≠ b < sub > j </ sub > are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m + n ≥ k.

For if k, m, and n are integers, and k is a common factor of two integers A and B, then A = nk and B = mk implies A − B = ( n − m ) k, therefore k is also a common factor of the difference.

There is a link between the " decision " and " optimization " problems in that if there exists a polynomial algorithm that solves the " decision " problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k.

On the other hand, if an algorithm finds the optimal value of optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k.

In operating systems, the execution of the executable can be postponed if other executables ( a. k. a. processes ) are also executing.

k < sub > 1 </ sub > + k < sub > 2 </ sub >-k < sub > 3 </ sub >, and so if the two pump beams are counterpropagating ( k < sub > 1 </ sub >

so it follows that if the voltage is doubled and the power remains the same, the current will be halved .</ ref >

if the item > largest, then

Antimony is stable in air at room temperature, but reacts with oxygen if heated to form antimony trioxide, Sb < sub > 2 </ sub > O < sub > 3 </ sub >.

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

The color of amethyst has been demonstrated to result from substitution by irradiation of trivalent iron ( Fe < sup > 3 +</ sup >) for silicon in the structure, in the presence of trace elements of large ionic radius, and, to a certain extent, the amethyst color can naturally result from displacement of transition elements even if the iron concentration is low.

The " Mask of Agamemnon " which was discovered by Heinrich Schliemann in 1876 at Mycenae ( whether it represents an individual, and if so, whom, remains unknown )</ center >

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

After the indecisive < ref name =" British historian Townsend Miller "> British historian Townsend Miller: “ But, if the outcome of < nowiki > battle of </ nowiki > Toro, militarily, is debatable, there is no doubt whatsoever as to its enormous psychological and political effects ” in The battle of Toro, 1476, in History Today, volume 14, 1964, p. 270 </ ref > Battle of Toro in 1476 against King Ferdinand II of Aragon, the husband of Isabella I of Castile, he went to France to obtain the assistance of Louis XI, but finding himself deceived by the French monarch, he returned to Portugal in 1477 in very low spirits.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.