The larger sieve is applied with the set \ mathcal { S } above defined to be the set of primes for which many residue classes are removed, while the large sieve is used to obtain information using the primes outside \ mathcal { S }.
32.
And for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates " i " such that " a " " i " belongs to this residue class, is a word in the binary Golay code.
33.
And for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates " i " such that " a " " i " belongs to this residue class, is a word in the binary Golay code.
34.
A simple case is that seen in the Gauss lemma on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number " p ", with respect to the subgroup { 1, & minus; 1 }.
35.
To prove ( i ), first note that we can compute g ( k ) either directly, i . e . by plugging in ( the residue class of ) k and performing arithmetic in \ textstyle \ mathbb { Z } / p, or by reducing f ( k ) \ mod p.
36.
Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and " c " and " f " can be read off from the valuations of " j " and ? ( defined below ).
37.
Both of the proofs ( for prime moduli ) below make use of the fact that the residue classes modulo a prime number are a Lagrange's theorem, which states that in any field a polynomial of degree " n " has at most " n " roots, is needed for both proofs.
38.
The splitting invariant of a prime " p " not dividing " N " is simply its residue class because the number of distinct primes into which " p " splits is ? ( " N " ) / m, where m is multiplicative order of " p " modulo " N; " hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to " N ".
39.
The splitting invariant of a prime " p " not dividing " N " is simply its residue class because the number of distinct primes into which " p " splits is ? ( " N " ) / m, where m is multiplicative order of " p " modulo " N; " hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to " N ".
40.
If we then lift ( mod 3 ) then we get two or four cases ( depending on whether either of " m " or " d " can be congruent to zero ( mod 3 ) ), and using the Chinese remainder theorem to glue these cases to the cases derived from " n " ( mod 16 ), we end up with four or eight cases-- some from residue classes ( mod 12 ) and some from residue classes ( mod 24 ) .)
