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The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the n × n {\displaystyle n\times n} matrix ( gcd ( i , j ) ) {\displaystyle (\gcd(i,j))} is ϕ ( 1 ) ϕ ( 2 ) ⋯ ϕ ( n ) {\displaystyle \phi (1)\phi (2)\cdots \phi (n ...
Therefore, equalities like d = gcd(p, q) or gcd(p, q) = gcd(r, s) are common abuses of notation which should be read "d is a GCD of p and q" and "p and q have the same set of GCDs as r and s". In particular, gcd( p , q ) = 1 means that the invertible constants are the only common divisors.
Since b ≥ φ N−1, then N − 1 ≤ log φ b. Since log 10 φ > 1/5, (N − 1)/5 < log 10 φ log φ b = log 10 b. Thus, N ≤ 5 log 10 b. Thus, the Euclidean algorithm always needs less than O divisions, where h is the number of digits in the smaller number b.
[1] A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions:
For a concrete example one can take R = Z[i√5], p = 1 + i√5, a = 1 − i√5, q = 2, b = 3. In this example the polynomial 3 + 2 X + 2 X 2 (obtained by dividing the right hand side by q = 2 ) provides an example of the failure of the irreducibility statement (it is irreducible over R , but reducible over its field of fractions Q [ i √5] ).
In number theory, the gcd-sum function, [1] also called Pillai's arithmetical function, [1] is defined for every by = = (,) or ...
The assumption is that the loop must be normalized – written so that the loop index/variable starts at 1 and gets incremented by 1 in every iteration. For example, in the following loop, a=2, b=3, c=2, d=0 and GCD(a,c)=2 and (d-b) is -3. Since 2 does not divide -3, no dependence is possible.
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