S428 200k<n<1M sieved to 2e12 (it was faster than I expected). At this score, I have ~125 seconds per factor, which would mean ~1000 seconds per factor in the 200k-300k range. Some cllr test in the same cpu says about 1800-2000 seconds for one test.

There are <7500 candidates remaining in 200k<n<1M.

**Question for the people knowing more about this subject: how much further should I go with sieving? And should I post the file here (~70kB) or send it to Gary?**
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In the same time I ran cllr in a different core for 8*428^n+1 for n to about 70k (continuing) - this is a doublecheck.

**Gary, do you need residue file for DC purposes?**
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I also ran low-sieving and cllr for all other k's (as there are only 9 all together) for n>=2, as I just didn't like those numbers being prime for n=1.

This is for the category "futile work", but it still can be posted like a puzzle. I found primes for n higher than 1 for all of them, except 2 and 6, but 6 is eliminated by algebraic factorization, it seems that all of them are divisible by 7. So here the primes with higher n:

Code:

2*428^1+1 = 857 is prime! (trial divisions)
5*428^1+1 = 2141 is prime! (trial divisions)
9*428^1+1 = 3853 is prime! (trial divisions)
3*428^2+1 = 549553 is prime! (trial divisions)
7*428^2+1 = 1282289 is prime! (trial divisions)
9*428^3+1 = 705624769 is prime! (trial divisions)
1*428^32+1 is prime! (85 decimal digits) Time : 9.273 ms.
3*428^15+1 is prime! (40 decimal digits) Time : 9.067 ms.
4*428^14+1 is prime! (38 decimal digits) Time : 13.575 ms.
5*428^21+1 is prime! (56 decimal digits) Time : 9.341 ms.
7*428^20+1 is prime! (54 decimal digits) Time : 9.544 ms.
9*428^1665+1 is prime! (4383 decimal digits) Time : 156.265 ms.

Now, the "puzzle" is that I didn't find any prime for k=2, although I sieved to 1T and I cllr it to n=69133 - still running.

(I am not so happy with the prime for k=4 also, as that is long known, and not a "new find", hehe, but I didn't do any action in that direction.

)

One outcome from this futile playing with numbers is that I uncovered a small bug in cllr. My first sieving file was including 10*428^n+1 too, and it was sieving from n=0, therefore after srsieve kept 10*428^0+1 (which is prime) and eliminated all the other, so in the resulted sieved file (I stopped srsieve at 1e8 and continued with sr2sieve, then split and used sr1sieve for 2 and 8) - so in resulted file after first application of the srsieve, the 2*428^1+1 (=857, prime) followed after 10*428^0+1 (=11, also prime). In this very particular situation, cllr is testing the first one, and says it's prime, but is skipping the other one (i.e. misses the prime 857). Now I know this is a buggy situation, which will never happen in real life, but I wonder which other primes it can miss...