######
§27.17 Other Applications

…

►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom

numbers.
…

►There are also applications of

number theory in many diverse areas, including physics, biology, chemistry, communications, and art.

Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

######
§27.16 Cryptography

…

►The primes are kept secret but their product

$n=pq$, an 800-digit

number, is made public.
…With the most efficient computer techniques devised to date (2010), factoring an 800-digit

number may require billions of years on a single computer.
For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large

numbers.
…

►For further information see

Apostol and Niven (1994, p. 24), and for other applications to cryptography see

Menezes et al. (1997) and

Schroeder (2006).