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nonrepresentable

Nonrepresentable is a term used in mathematics and related fields to describe objects that cannot be represented within a given representation system or form. The precise meaning depends on the chosen context or encoding, and the term is often used to distinguish what is possible to express under a specified set of rules from what is not.

In number theory, nonrepresentability is commonly discussed in relation to expressing integers by a fixed form.

Nonrepresentability can also apply to representations under other constraints or forms, such as linear combinations with

See also: representable, representation theory, encoding, computability.

A
standard
example
is
representation
as
a
sum
of
two
squares.
An
integer
n
is
expressible
as
a^2
+
b^2
for
some
integers
a
and
b
if
and
only
if
every
prime
congruent
to
3
modulo
4
occurs
in
the
prime
factorization
of
n
with
even
exponent.
Consequently,
many
numbers
are
nonrepresentable
in
this
form;
for
instance,
3,
6,
7,
11,
14,
15,
19,
21,
and
22
are
not
sums
of
two
squares,
while
numbers
such
as
0,
1,
2,
4,
5,
8,
9,
and
10
are
representable.
restricted
coefficients,
or
encoding
within
a
particular
formal
system.
In
logic
and
computer
science,
a
value
or
function
may
be
termed
nonrepresentable
relative
to
a
given
encoding,
automaton,
or
language
if
no
valid
representation
exists
within
that
system.