Algorithm for finding the longest prefix
I am assuming data type text for the relevant columns.

CREATE TABLE prefix (code text, name text, price int);
CREATE TABLE num (number text, time int);


"Simple" Solution

SELECT DISTINCT ON (1)
       n.number, p.code
FROM   num n
JOIN   prefix p ON right(n.number, -1) LIKE (p.code || '%')
ORDER  BY n.number, p.code DESC;


Key elements:

DISTINCT ON is a Postgres extension of the SQL standard DISTINCT. Find a detailed explanation for the used query technique in this related answer on SO.
ORDER BY p.code DESC picks the longest match, because '1234' sorts after '123' (in ascending order).

Simple SQL Fiddle.

Without index, the query would run for a very long time (didn't wait to see it finish). To make this fast, you need index support. The trigram indexes you mentioned, supplied by the additional module pg_trgm are a good candidate. You have to choose between GIN and GiST index. The first character of the numbers is just noise and can be excluded from the index, making it a functional index in addition.
In my tests, a functional trigram GIN index won the race over a trigram GiST index (as expected):

CREATE INDEX num_trgm_gin_idx ON num USING gin (right(number, -1) gin_trgm_ops);


Advanced dbfiddle here.

All test results are from a local Postgres 9.1 test installation with a reduced setup: 17k numbers and 2k codes:


Total runtime: 1719.552 ms (trigram GiST)
Total runtime: 912.329 ms (trigram GIN)


Much faster yet

Failed attempt with text_pattern_ops

Once we ignore the distracting first noise character, it comes down to basic left anchored pattern match. Therefore I tried a functional B-tree index with the  operator class text_pattern_ops (assuming column type text).

CREATE INDEX num_text_pattern_idx ON num(right(number, -1) text_pattern_ops);


This works excellently for direct queries with a single search term and makes the trigram index look bad in comparison:

SELECT * FROM num WHERE right(number, -1) LIKE '2345%'



Total runtime: 3.816 ms (trgm_gin_idx)
Total runtime: 0.147 ms (text_pattern_idx)


However, the query planner will not consider this index for joining two tables. I have seen this limitation before. I don't have a meaningful explanation for this, yet.

Partial / functional B-tree indexes

The alternative it to use equality checks on partial strings with partial indexes. This can be used in a JOIN.

Since we typically only have a limited number of different lengths for prefixes, we can build a solution similar to the one presented here with partial indexes.

Say, we have prefixes ranging from 1 to 5 characters. Create a number of partial functional indexes, one for every distinct prefix length:

CREATE INDEX prefix_code_idx5 ON prefix(code) WHERE length(code) = 5;
CREATE INDEX prefix_code_idx4 ON prefix(code) WHERE length(code) = 4;
CREATE INDEX prefix_code_idx3 ON prefix(code) WHERE length(code) = 3;
CREATE INDEX prefix_code_idx2 ON prefix(code) WHERE length(code) = 2;
CREATE INDEX prefix_code_idx1 ON prefix(code) WHERE length(code) = 1;


Since these are partial indexes, all of them together are barely larger than a single complete index.

Add matching indexes for numbers (taking the leading noise character into account):

CREATE INDEX num_number_idx5 ON num(substring(number, 2, 5)) WHERE length(number) &gt;= 6;
CREATE INDEX num_number_idx4 ON num(substring(number, 2, 4)) WHERE length(number) &gt;= 5;
CREATE INDEX num_number_idx3 ON num(substring(number, 2, 3)) WHERE length(number) &gt;= 4;
CREATE INDEX num_number_idx2 ON num(substring(number, 2, 2)) WHERE length(number) &gt;= 3;
CREATE INDEX num_number_idx1 ON num(substring(number, 2, 1)) WHERE length(number) &gt;= 2;


While these indexes only hold a substring each and are partial, each covers most or all of the table. So they are much larger together than a single total index - except for long numbers. And they impose more work for write operations. That's the cost for amazing speed.

If that cost is too high for you (write performance is important / too many write operations / disk space an issue), you can skip these indexes. The rest is still faster, if not quite as fast as it could be ...

If numbers are never shorter then n characters, drop redundant WHERE clauses from some or all, and also drop the corresponding WHERE clause from all following queries.

Recursive CTE

With all the setup so far I was hoping for very elegant solution with a recursive CTE:

WITH RECURSIVE cte AS (
   SELECT n.number, p.code, 4 AS len
   FROM   num n
   LEFT    JOIN prefix p
            ON  substring(number, 2, 5) = p.code
            AND length(n.number) &gt;= 6  -- incl. noise character
            AND length(p.code) = 5

   UNION ALL 
   SELECT c.number, p.code, len - 1
   FROM    cte c
   LEFT   JOIN prefix p
            ON  substring(number, 2, c.len) = p.code
            AND length(c.number) &gt;= c.len+1  -- incl. noise character
            AND length(p.code) = c.len
   WHERE    c.len &gt; 0
   AND    c.code IS NULL
   )
SELECT number, code
FROM   cte
WHERE  code IS NOT NULL;



Total runtime: 1045.115 ms


However, while this query isn't bad - it performs about as good as the simple version with a trigram GIN index - it doesn't deliver what I was aiming for. The recursive term is planned once only, so it can't use the best indexes. Only the non-recursive term can.

UNION ALL

Since we are dealing with a small number of recursions we can just spell them out iteratively. This allows optimized plans for each of them. (We lose the recursive exclusion of already successful numbers, though. So there is still some room for improvement, especially for a wider range of prefix lengths)):

SELECT DISTINCT ON (1) number, code
FROM  (
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 5) = p.code
            AND length(n.number) &gt;= 6  -- incl. noise character
            AND length(p.code) = 5
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 4) = p.code
            AND length(n.number) &gt;= 5
            AND length(p.code) = 4
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 3) = p.code
            AND length(n.number) &gt;= 4
            AND length(p.code) = 3
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 2) = p.code
            AND length(n.number) &gt;= 3
            AND length(p.code) = 2
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 1) = p.code
            AND length(n.number) &gt;= 2
            AND length(p.code) = 1
   ) x
ORDER BY number, code DESC;



Total runtime: 57.578 ms (!!)


A breakthrough, finally!

SQL function

Wrapping this into an SQL function removes the query planning overhead for repeated use:

CREATE OR REPLACE FUNCTION f_longest_prefix()
  RETURNS TABLE (number text, code text) LANGUAGE sql AS
$func$
SELECT DISTINCT ON (1) number, code
FROM  (
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 5) = p.code
            AND length(n.number) &gt;= 6  -- incl. noise character
            AND length(p.code) = 5
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 4) = p.code
            AND length(n.number) &gt;= 5
            AND length(p.code) = 4
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 3) = p.code
            AND length(n.number) &gt;= 4
            AND length(p.code) = 3
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 2) = p.code
            AND length(n.number) &gt;= 3
            AND length(p.code) = 2
   UNION ALL 
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(number, 2, 1) = p.code
            AND length(n.number) &gt;= 2
            AND length(p.code) = 1
   ) x
ORDER BY number, code DESC
$func$;


Call:

SELECT * FROM f_longest_prefix_sql();



Total runtime: 17.138 ms (!!!)


PL/pgSQL function with dynamic SQL

This plpgsql function is much like the recursive CTE above, but the dynamic SQL with EXECUTE forces the query to be re-planned for every iteration. Now it makes use of all the tailored indexes.

Additionally this works for any range of prefix lengths. The function takes two parameters for the range, but I prepared it with DEFAULT values, so it works without explicit parameters, too:

CREATE OR REPLACE FUNCTION f_longest_prefix2(_min int = 1, _max int = 5)
  RETURNS TABLE (number text, code text) LANGUAGE plpgsql AS
$func$
BEGIN
FOR i IN REVERSE _max .. _min LOOP  -- longer matches first
   RETURN QUERY EXECUTE '
   SELECT n.number, p.code
   FROM   num n
   JOIN   prefix p
            ON  substring(n.number, 2, $1) = p.code
            AND length(n.number) &gt;= $1+1  -- incl. noise character
            AND length(p.code) = $1'
   USING i;
END LOOP;
END
$func$;


The final step cannot be wrapped into the one function easily.
Either just call it like this:

SELECT DISTINCT ON (1)
       number, code
FROM   f_longest_prefix_prefix2() x
ORDER  BY number, code DESC;



Total runtime: 27.413 ms


Or use another SQL function as wrapper:

CREATE OR REPLACE FUNCTION f_longest_prefix3(_min int = 1, _max int = 5)
  RETURNS TABLE (number text, code text) LANGUAGE sql AS
$func$
SELECT DISTINCT ON (1)
       number, code
FROM   f_longest_prefix_prefix2($1, $2) x
ORDER  BY number, code DESC
$func$;


Call:

SELECT * FROM f_longest_prefix3();



Total runtime: 37.622 ms


A bit slower due to required planning overhead. But more versatile than SQL and shorter for longer prefixes.