Prime Numbers

Appendix BOOK PSEUDOCODE 543
t = x; //Variable t is temporary here.
x = y 2 ;
y =2t;
and it is quite evident by comparison how visually efficient is the single-line
vector assignment. Note, too, that the composite assignments
and
x = y 2 ;
y =2x;
y =2x;
x = y 2 ;
are both different than the vector assignment, and different from each other.
Because our text adheres to the rule that ordered sequences are symbolized
by parentheses (as in (xn)) while sets use braces (as in {X, a, α}), we assign
sequences, vectors, and so on with a style consistent with the text; e.g.,
v =(0, 1, 0) is an ordered assignment, whereas a set of three polynomials
might be assigned as S = {x 2 +1,x,x 3 − x} and the order is unimportant.
Moreover, S = {x 2 +1,x,x,x 3 − x} is exactly the same assignment, since
set notation does not record multiplicity. Note that the distinction between
sequence and set assignment is important, in view of the liberal use of braces
in modern languages. In the Mathematica language, braces denote “lists” and
these in turn can be manipulated as either vectors (sequences) or sets, with
vector-algebraic (such as matrix-vector) and set-theoretical (such as union,
intersection) operators available. Likewise, the C language allows assignment
of data records via braces, as in “float x[3] = {1.1, 2.2, 3.3};” which would fill
a vector x in ordered fashion. In this latter case, our pseudocode would say
instead x =(1.1, 2.2, 3.3).
The internal conditionals in if() statements often use classical mathematical
notation, but not always. Let us exemplify conditional syntax like so:
if(x == y) task(); // Testing equality of x, y, without changing either.
if(x ≥ y) task(); // Testing whether x is greater than or equal to y.
if(x|y) task(); // Testing whether x divides y.
if(x ≡ y (mod p)) task(); // Testing whether x, y congruent modulo p.
Note that a congruence conditional does not take the form x ≡≡ y (mod p),
because there need not be any confusion with assignment in such cases.
However, it may be possible to have the construction x == y mod p, sinceas
is explained in the text, the notation y mod p refers to the integer y − p ⌊y/p⌋.
Thus, it may be that x is equal to this integer, or it may be that we wish to
assign x this value (in which case we would write x = y mod p).
Another conditional form is the while() statement, exemplified by
while(x = 0) {
task1();
task2();