Pocklington's algorithm is a technique for solving a
congruence
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
of the form
:
where ''x'' and ''a'' are integers and ''a'' is a
quadratic residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic no ...
.
The algorithm is one of the first efficient methods to solve such a congruence. It was described by
H.C. Pocklington in 1917.
[H.C. Pocklington, Proceedings of the Cambridge Philosophical Society, Volume 19, pages 57–58]
The algorithm
(Note: all
are taken to mean
, unless indicated otherwise.)
Inputs:
* ''p'', an odd
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
* ''a'', an integer which is a quadratic residue
.
Outputs:
* ''x'', an integer satisfying
. Note that if ''x'' is a solution, −''x'' is a solution as well and since ''p'' is odd,
. So there is always a second solution when one is found.
Solution method
Pocklington separates 3 different cases for ''p'':
The first case, if
, with
, the solution is
.
The second case, if
, with
and
#
, the solution is
.
#
, 2 is a (quadratic) non-residue so
. This means that
so
is a solution of
. Hence
or, if ''y'' is odd,
.
The third case, if
, put
, so the equation to solve becomes
. Now find by trial and error
and
so that
is a quadratic non-residue. Furthermore, let
:
.
The following equalities now hold:
:
.
Supposing that ''p'' is of the form
(which is true if ''p'' is of the form
), ''D'' is a quadratic residue and
. Now the equations
:
give a solution
.
Let
. Then
. This means that either
or
is divisible by ''p''. If it is
, put
and proceed similarly with
. Not every
is divisible by ''p'', for
is not. The case
with ''m'' odd is impossible, because
holds and this would mean that
is congruent to a quadratic non-residue, which is a contradiction. So this loop stops when
for a particular ''l''. This gives
, and because
is a quadratic residue, ''l'' must be even. Put
. Then
. So the solution of
is got by solving the linear congruence
.
Examples
The following are 4 examples, corresponding to the 3 different cases in which Pocklington divided forms of ''p''. All
are taken with the
modulus in the example.
Example 0
:
This is the first case, according to the algorithm,
, but then
not 43, so we should not apply the algorithm at all. The reason why the algorithm is not applicable is that a=43 is a quadratic non residue for p=47.
Example 1
Solve the congruence
:
The modulus is 23. This is
, so
. The solution should be
, which is indeed true:
.
Example 2
Solve the congruence
:
The modulus is 13. This is
, so
. Now verifying
. So the solution is
. This is indeed true:
.
Example 3
Solve the congruence
. For this, write
. First find a
and
such that
is a quadratic nonresidue. Take for example
. Now find
,
by computing
:
:
And similarly
such that
Since
, the equation
which leads to solving the equation
. This has solution
. Indeed,
.
References
* Leonard Eugene Dickson, "History Of The Theory Of Numbers" vol 1 p 222, Chelsea Publishing 1952
{{number theoretic algorithms
Modular arithmetic
Number theoretic algorithms