Relatively prime

It is easy to show that every polynomial f x) (not divisible by x) over a finite field J g is a factor of 1—cc , for some power n . The order (sometimes also called the period or exponent) of f x), denoted by ord(/), is the least such n . If f x) = p x) is an irreducible polynomial (other than x) with d[f] = n then ord(/) must divide pTi i There are two important theorems concerning the orders of prime factors and products of relatively prime polynomials over Fg ... 

SIMPLIFIED FRACTION a fraction whose numerator and denominator are relatively prime... 

RELATIVELY PRIME any numbers whose only common factor is 1... 

The Horiuti numbers defining the cyclic characteristic in Equation (34) are relatively prime i.e. GCD(vi,..., v ) — 1. The exponent p in Equation (34) is the natural number. If we assume additionally that... 

Remind that v belongs to the set of relatively prime stoichiometric numbers, so that... 

The tip of the typical resonance horn touches the axis FA = 0 at points )/ >o = p/q, where p and q are relatively prime integers. The periodic trajectories within such a horn have a period pT, where T is the forcing period, and we refer to them as p-periodic trajectories or simply period ps. The larger the relatively prime numbers p and q, the narrower the resonance horn and the sharper its tip. 

A right triangle whose sides are relatively prime integers, primitive root of unity... 

For a simple (primitive) cubic lattice P, there are no restrictions on the Miller indices and therefore none on KF except as noted above. In the case of a body-centered cubic lattice, only those values of KF can be allowed that arise from Miller indices whose sum is even. This has the effect of requiring KF to be even, as seen in the first of the two columns under /.We can then divide them by 2 and thereby reduce them to a relatively prime sequence, shown in the second column under /, for comparison with the sequence obtained from the ldf values. We note immediately that the relatively prime sequence obtained differs from that for a primitive cubic lattice in having gaps at different places. By use of this fact, it is almost always possible to distinguish between a primitive cubic lattice and a... 

If no relatively prime sequence of integers can be found to within the experimental uncertainty of the measurements, the crystalline substance presumably does not belong to the cubic system. 

Find a factor that will reduce the Hdf values to relatively prime integers within experimental error. Refer to Table 1 and identify the lattice type. Also from Table 1, obtain the Miller indices hkl for each fine. 

Miller indices, hkl Three relatively prime integers, hkl, that are reciprocals of the fractional intercepts that the crystallographic plane makes with the crystallographic axes. The crystallographic plane hkl) is described by its Miller indices. 

There axe members of the number continuum that are different from rational. The number y/2, e.g., is not rational. The proof was known to the ancient Greeks assume a/2 is rational. Then, it can be written as /2 = n/m where n and m are integers and relatively prime. Squaring this relation we get = 2m. This means that contains a factor 2. But if contains a factor 2, so does n. Therefore, we write = 4p, where p is another integer. Thus, = 2p. Using the same argument as before, m contains a factor 2. But this contradicts the assumption that n and m are relatively prime. Thus we proved that is not rational. 

The parameterization chooses a set of additive constants bj that are pairwise relatively prime, gcd(b , bj) — 1, when i j so that the sequences are generated in different orders. The best choice is to let bj be the jth prime less than /2 [14]. One important advantage of this parameterization is that there is an interstream correlation measure based on the spectral test that suggests that there will be good interstream independence. 

Definition 2.2.8. 7Wo integers x and y are relatively prime if there is no positive integer greater than 1 that divides them both. 

The prime factorizations of the positive integers x and y are useful in detennining whether x and y are relatively prime. For Example 2.2.9, in the second case, it was determined that 6 and 14 are not relatively prime. Given their prime factorizations, 6 = 2-3 and 14 = 2 7, it can readily be seen that 2 divides both numbers and thus that 6 and 14 are not relatively prime. In the first case presented in Example 2.2.9, it was determined that 6 and 35 are relatively printe. Given their prime factorizations, 6 2-3 and 35 = 5-7,... 

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