**Solution:**

**Faride Firoozbakht** found the following
results:

__f(x), f(m) is composite for all m up to__

x^6 + 801967256, 28958

x^6 + 907906544, 33914

x^6 + 995925839, 29399

x^6 + 1209089636, 28118

x^6 + 1225641416, **34964**

Later she added:

The largest value of k that I found for c < 0 is
39458; and for c > 0 is 36350, the corresponding polynomials are in the
following table.

f(x) |f(m)| is composite for all m up to

x^6 - 2335031854 39458

x^6 + 2946524426 36350

***

The 31 of July (2004) **Adam Stinchcombe**
wrote:

One approach proved very fruitful. Starting with
the idea that c0=5983049981 was a good value for c, i.e., it gave useful
arithmetic properties mod various primes, namely one can solve x^6+c0=0
mod p for each p<=29, thus x^6+c0 will be large in size and divisible by
small primes, and so therefore composite, for most values of x. I then
used a modulus of m=29#=6469693230, calculated ci=c0+m*i and came up with
c=32386797666131, # of composites = 30323 and then c=432129733268141, # of
composites = 53255

***

**Phil Carmody** got a new record for this puzzle
and at the same time four larger examples than the gotten by **
Stinchcombe**:

It looked like 30000 was easy to achieve, but the
40000s were pretty rare, and 50000s were absolutely impossible.

Here are the ones I found over 39458:

1292597109886991 46956

2683424369451671 39900

5436756742672646 40845

11117808561725774 45297

17577986970290501 40404

20726454730694621 41160

24164653747817519 42588

**29450922301244534 63693**

28715053738503404 51597

30059408353405976 45885

**34964348452640606 53865**

38631587821244354 56343

45223406878461356 53529

As you see, 50000 was impossible, until the 60000
removed whatever blockage was in my program :-). Note that large c makes
it easier, so really there should be some weighting (maybe divide by
log(c)) to make up for this.

***

On October 31, 2016, Dmitry Kamenetsky wrote:

x^12+488669 is composite for 0<x<616980, see http://oeis.org/A122131.