Nothing of what I am about to write is new. Or difficult.
And it has already been discussed multiple times.

Observe that some the chosen bases for this particular sub-project happen to be squares.
25 = 5^2, 49 = 7^2, 121 = 11^2. Some people even know that this was a deliberate choice. I've been waiting for a workunit to arrive for one of these bases for a while, and now I have one.

The candidate is 754806*121^754806+1 which can also be easily regrouped as 754806*11^1509612+1.

Let's compare:

/home/serge/NumTheory/GCW> llr -d d.npg
Base prime factor(s) taken : 11
Starting N-1 prime test of 754806*121^754806+1
Using zero-padded AVX FFT length 720K, Pass1=320, Pass2=2304, a = 3
754806*121^754806+1, bit: 70000 / 5222413 [1.34%]. Time per bit: 5.654 ms.

/home/serge/NumTheory/GCW/2> llr -d d2.npg
Base prime factor(s) taken : 11
Starting N-1 prime test of 754806*11^1509612+1
Using zero-padded AVX FFT length 640K, Pass1=640, Pass2=1K, a = 3
754806*11^1509612+1, bit: 40000 / 5222416 [0.76%]. Time per bit: 4.698 ms.

For LLR it is not the number that matters, but the (k,b,n,c) form, and b is taken by it verbatim, as given.

I am not ready to go into a very deep explanation, but I will try to make an approximation to an explanation (not meant to be taken for that this is exact). With b=11, it is possible to form an array of length 640K where each element is a quasi-digit ("limb") in an unusual representation: some digit's weights are perhaps 11^6 and some digit's weights are 11^7, or something like that. (Off the top of my head, what I remember is that each limb on average is limited to keeping ~30 bits of information, or something like that.) Only powers of b can be used as limb weights.

In contrast if the number is entered with b=121, the program can only work with limbs of, say, 121^3 and 121^2. (It has less opportunities to pack, the larger the b.) For that reason it ponders the array of length 640K and thinks, "nah, some elements will be too large; gotta go for next FFT size", and so it does.

Long story short, using the simplest possible (k,b,n,c) (with b as low as possible) will lend more possibilities for more dense FFT arrays. If not only b=121, but also k is divisible by 11, the FFT size may be even smaller if (k,121,n,c) is transformed into (k/11,11,2*n+1,c).

As an aside, yes, it would be nice if LLR did it all itself, but as the timing test (shown earlier) demonstrates - it doesn't. But here's where we can help LLR and do transformation externally, server-side. The transformation logic is quite straightforward.[/i]

Observe that some the chosen bases for this particular sub-project happen to be squares.
25 = 5^2, 49 = 7^2, 121 = 11^2. Some people even know that this was a deliberate choice.

It seems that in your client-server set up the ideal place to put the reformatter would be the primegrid_llr_wrapper. It would keep the initial task parameters, reformat for (c)llr, get the result back from (c)llr, report back to server as initially requested. Then the database, the server and the accounting code would be unchanged.

primegrid_llr_wrapper for now can do only:
▪ square simplification,
▪ k simplification

Later, it can be extended to recognize b being any power. See here -

Curiously, these numbers may be hard to recognize when written in standard form (emphasis mine).

For example, they may be like
18740*3^168662-1
which could be written
168660*3^168660-1.

More difficult to spot are those like the following:

9750*7^29250-1 = 9750*7^(3*9750)-1 = 9750*343^9750-1
8511*2^374486-1 = (8511*2^2)*2^(11*8511)*4-1 = 34044*2048^34044-1.

Maybe LLR's philosophy is "the client is always right!"
I.e.: If the input file calls for a test of a specific FFT or a "specific arrangement of bits", then that's what it will run (even if slower, because "this is the test that was ordered").

But it indeed doesn't follow this rule for powers of 2.

-bash-4.2$ llr -d -q"27*1024^10007+1"
Starting Proth prime test of 27*2^100070+1
Using all-complex FMA3 FFT length 10K, Pass1=128, Pass2=80, a = 11
27*2^100070+1 is not prime. Proth RES64: C14E6737D2E78E5E Time : 5.261 sec.

-bash-4.2$ llr -d -q"28*729^10007+1"
Base prime factor(s) taken : 3
Starting N-1 prime test of 28*729^10007+1
Using all-complex FMA3 FFT length 10K, Pass1=128, Pass2=80, a = 3
28*729^10007+1 is not prime. RES64: E83080E955E9B281. OLD64: B89182BC01BD1780 Time : 4.888 sec.

-bash-4.2$ llr -d -q"28*10000^10007+1"
Base factorized as : 2^4*5^4
Base prime factor(s) taken : 5
Starting N-1 prime test of 28*10000^10007+1
Using all-complex FMA3 FFT length 18K, Pass1=384, Pass2=48, a = 3
28*10000^10007+1 is not prime. RES64: 59CCA66A39ED54C4. OLD64: 0D65F33EADC7FE48 Time : 13.645 sec.

And GeneFer seems to do different things, not normalizing or de-normalizing:

.\genefer_windows64.exe -q "6^8388608+1"
.\genefer_windows64.exe -q "36^4194304+1"
.\genefer_windows64.exe -q "1296^2097152+1"
.\genefer_windows64.exe -q "1679616^1048576+1"

Even though the first form (where b=6 is not a square) is "canonical" and the one you would expect to see on Top 5000, it is not clear which form would actually be fastest.

Testing 6^8388608+1... 21684224 steps to go (1849:28:44 remaining)

Testing 36^4194304+1... 21684224 steps to go (747:08:06 remaining)

Testing 1296^2097152+1... 21684224 steps to go (367:41:08 remaining)

Testing 1679616^1048576+1... 21684224 steps to go (160:41:45 remaining)
Estimated time remaining for 1679616^1048576+1 is 1716:50:53

(the last one is switches to x87 (80-bit) transform).

/JeppeSN

I could've sworn that LLR itself does the normalizing of the bases (perhaps only for power of 2?). This feature needs to be in LLR itself, tbh.