factor ---debug output is rather rudimentary.
In order to understand the logic, I added more printouts
(try attached debug patch)
and immediately noticed some obvious snafus:

$ src/factor ---debug 340282366920938461286658806734041124249
[using arbitrary-precision arithmetic] 340282366920938461286658806734041124249:[trial division 340282366920938461286658806734041124249] [trial stopped 4999(max)] [is number 34028236692093846128665880673404112424
9 prime?] [no] [pollard-rho (1)] 1:1:2:4:8:16:32:64:128:256:512:1024:2048:4096:8192:16384:32768:65536:131072:262144:524288:1048576:2097152:4194304:8388608:16777216:33554432:67108864:134217728:268435456:536870912
[factor found][rho factor 18446744073709551557] [trial division 18446744073709551556] [factor 2] [factor 2] [factor 11] [factor 137] [factor 547] [trial stopped 4999(max)] [is number 5594472617641 prime?] [trial
  division 5594472617640] [factor 2] [factor 2] [factor 2] [factor 3] [factor 5] [factor 1427] [factor 2131] [trial stopped 2131] [is number 15331 prime?] [yes] [factor 15331] [yes] ...

Look at the ens of the last line.
We don't need to check whether 15331 is prime. We know it is: we just stopped
trial division because next prime divisor to try is larger than
sqrt(number_we_try_to_factor) - therefore number_we_try_to_factor
has no more divisors, ergo it's prime. The fix:

    for (unsigned int i = 1; i <= PRIMES_PTAB_ENTRIES;)
      {
        if (! mpz_divisible_ui_p (t, p))
          {
            p += primes_diff[i++];
            if (mpz_cmp_ui (t, p * p) < 0)
+            mp_factor_insert (factors, t);
+            mpz_set_ui (t, 1);
              break;
          }
        else


... [factor 5594472617641] [factor
  18446744073709551557] [trial division 18446744073709551556] [factor 2] [factor 2] [factor 11] [factor 137] [factor 547] [trial stopped 4999(max)] [is number 5594472617641 prime?] [trial division 5594472617640]
[factor 2] [factor 2] [factor 2] [factor 3] [factor 5] [factor 1427] [factor 2131] [trial stopped 2131] [is number 15331 prime?] [yes] [factor 15331] [yes] [factor 5594472617641] [factor 18446744073709551557] [p
ollard-rho end]
   18446744073709551557 18446744073709551557

Second optimization is this: as you see, I deliberately fed it a square
(the factorization result is 18446744073709551557^2).
It happily did not notice it and spent an awful lot of time in pollard rho:

  1:1:2:4:8:16:32:64:128:256:512:1024:2048:4096:8192:16384:32768:65536:131072:262144:524288:1048576:2097152:4194304:8388608:16777216:33554432:67108864:134217728:268435456:536870912

A quick check for square drastically cuts down execution time:


  /* Use Pollard-rho to compute the prime factors of
     arbitrary-precision T, and put the results in FACTORS.  */
  static void
+mp_factor_big (mpz_t t, struct mp_factors *factors)
+{
+  /* the _big() function assumes trial division was already tried */
+  devmsg ("[is number prime?] ");
+  if (mp_prime_p (t))
+    mp_factor_insert (factors, t);
+  else
+    if (mpz_perfect_square_p (t))
+      {
+        struct mp_factors f2;
+        devmsg ("[it is square] ");
+        mpz_sqrt (t, t);
+        mp_factor_init (&f2);
+        mp_factor_big (t, &f2);
+        for (unsigned long int i = 0; i < f2.nfactors; i++)
+          {
+            unsigned long int e = f2.e[i];
+            while (e--) {                           //
+              mp_factor_insert (factors, f2.p[i]);  //FIXME: obvious optimization:
+              mp_factor_insert (factors, f2.p[i]);  // need to introduce/use mp_factor_insert_multiplicity(..., e*2);
+            }
+          }
+        mp_factor_clear (&f2);
+      }
+    else
+      mp_factor_using_pollard_rho (t, 1, factors);
+}
+
+static void
  mp_factor (mpz_t t, struct mp_factors *factors)
  {
    mp_factor_init (factors);
@@ -2265,11 +2309,7 @@ mp_factor (mpz_t t, struct mp_factors *f
.
        if (mpz_cmp_ui (t, 1) != 0)
          {
-          devmsg ("[is number prime?] ");
-          if (mp_prime_p (t))
-            mp_factor_insert (factors, t);
-          else
-            mp_factor_using_pollard_rho (t, 1, factors);
+          mp_factor_big (t, factors);
          }
      }
  }