The output of the integer operation determines a residue class, and the output of the modular operation is determined by computing the residue class's representative.
12.
The output of the integer operation determines a residue class, and the output of the modular operation is determined by computing the residue class's representative.
13.
A basic distinction is between a "'complete exponential sum "', which is typically a sum over all residue classes " Gauss ).
14.
There is a similar construction with "'Z " "'p " replaced by any complete discrete valuation ring with finite residue class field.
15.
We deduce that any periodic function with period " n " supported on the residue classes prime to " n " is a linear combination of Dirichlet characters.
16.
It is clear that the least residue system is a complete residue system, and that a complete residue system is simply a set containing precisely one representative of each residue class modulo.
17.
In terms of groups, the residue class } } is the coset of in the quotient group \ mathbb { Z } / n \ mathbb { Z }, a cyclic group.
18.
Tur�n was very interested in the distribution of primes in arithmetic progressions, and he coined the term " prime number race " for irregularities in the distribution of prime numbers among residue classes.
19.
Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers.
20.
Here there are good estimates for sums over shorter ranges than the whole set of residue classes, because, in geometric terms, the partial sums approximate a Cornu spiral; this implies massive cancellation.
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