Well…. The numbers after “12,298+ SNFS 323” in this table: https://escatter11.fullerton.edu/nfs/numbers.php will be

5,461+ SNFS 323
3,677- SNFS 324
5,463+ SNFS 324
7,383- SNFS 324
10,323- SNFS 324
10,323+ SNFS 324
11,311- SNFS 324
11,311+ SNFS 324
5,464+ SNFS 325
11,313- SNFS 326
5,467+ SNFS 327
6,419+ SNFS 327
7,386+ SNFS 327
6,421- SNFS 328
6,421+ SNFS 328
7,388+ SNFS 328
2,1091+ SNFS 329
7,389- SNFS 329
3,691- SNFS 330

Is it right? Now I known that the SNFS difficulty of b^n+-1 can be reduced to Phi(n,b) instead of b^n only if n has factors of 3, 5, 7, or 11, so for example, 7,395- and 7,395+ has difficulty 264, 10^371-1 has difficulty 312.

300 7 889 L 321.9 0.93
310 7 889 M 321.9 0.961
291 5 461 + 322.2 0.9
289 11 341 - 322.8 0.89 /11
259 3 677 - 323 0.8
271 10 323 - 323 0.839
242 10 323 + 323 0.749
268 2 1253 - 323.3 0.827 /7
280 3 791 - 323.4 0.864 /7
274 7 383 - 323.6 0.846
249 5 463 + 323.6 0.76
251 11 311 - 323.8 0.99
247 11 311 + 323.8 0.76

Thanks!!

I missed some large n-values which are 7*prime, especially 7,889L and 7,889M

also 11,341-, 341 is 11*prime

thus they have smaller difficulty.

It appears that 5*prime is not suitable for this, or 7,395-, 7,395+, 10,365-, 10,365+, etc. will be already factored in last year, what’s the reason?

5*prime should be also suitable, see https://www.rieselprime.de/ziki/SNFS_polynomial_selection

Thus 7,395- and 7,395+ should be SNFS 264, and 10,365- and 10,365+ should be SNFS 289, like that 10,371- is SNFS 313, and 3,791- is SNFS 324

So 7,395-, 7,395+, 10,365-, 10,365+ should be already factored, but they are not ….

Exponents divisible by 5 are possible but yield a degree-4 algebraic polynomial which is far from optimal at this size. They are more difficult than their SNFS-size suggests. See this paper for details. To a lesser extent the same applies for exponents divisible by 11, which results in a degree-5 algebraic polynomial.

I forgot to remove degree 5.
In 2021, NFS@Home factored two large degree 5 numbers, 6^451-1 (snfs 319), 3^737+1 (snfs 320) .
Here is a full list that snfs difficulty less than 330.
300 7 889 L 321.9 0.93 reserved
310 7 889 M 321.9 0.961 reserved
291 5 461 + 322.2 0.9 reserved
259 3 677 - 323 0.8 reserved
271 10 323 - 323 0.839 reserved
242 10 323 + 323 0.749 reserved
268 2 1253 - 323.3 0.827 /7 reserved
280 3 791 - 323.4 0.864 /7 reserved
274 7 383 - 323.6 0.846 reserved
249 5 463 + 323.6 0.76 reserved
251 11 311 - 323.8 0.99 reserved
247 11 311 + 323.8 0.76
223 5 464 + 324.3 0.68
256 7 416 + 324.5 0.787 /13
303 7 448 + 324.5 0.932 /7
281 11 338 + 324.9 0.86 /13
256 3 683 + 325.8 0.784
273 11 313 - 325.9 0.83
261 6 419 + 326 0.798
245 7 386 + 326.2 0.751
290 5 467 + 326.4 0.88
281 6 421 - 327.6 0.856
246 6 421 + 327.6 0.749
314 7 388 + 327.8 0.957
307 2 1091 + 328.4 0.934
299 7 389 - 328.7 0.909
252 3 805 + 329.2 0.763 /7
265 3 691 - 329.6 0.802
317 5 472 + 329.9 0.96

So only when exponent n is divisible by 3, 7, or 13, b^n+-1 has difficulty eulerphi(2*n)*log(b)/log(10) and is easier to factor? Otherwise (e.g. n is prime or 5*prime or 11*prime), b^n+-1 has difficulty n*log(b)/log(10) (more difficult than the SNFS difficulty when n is 5*prime or 11*prime)?

So does the status of numbers list https://escatter11.fullerton.edu/nfs/numbers.php now includes all numbers with SNFS difficulty <= 324?

How about 3,715-? 715 = 5*11*13, and the SNFS difficulty is 272.913, but as you say, exponents divisible by 5 or 11 is more difficult than their SNFS-size suggests, but exponents divisible by 3, 7, or 13 is not.

Can you reserve all remain numbers with difficulty < 325? (i.e. 11,311+ 5,464+ 7,416+7,448+ 11,338+)? (7,448+ is reserved by Dodson for 6 years with no result)

Now three numbers with exponents divisible by 5 (but not 3,7,13) are reserved, so what are their true difficulty? For 2,2350M, the SNFS suggestion difficulty is 283, and its true difficulty seems to be 324 (since it is reserved at this time, just after 11,311-, whose SNFS difficulty is 324), also 2,1180+ and 2,2390L, the SNFS suggestion difficulty are 285 and 288, respectively (also, why not reserve 7,395-? It is also exponent divisible by 5 and SNFS difficulty is 267, less than 2,2350M’s 283)

Can you reserve all remain numbers with difficulty < 325? (i.e. 11,311+ 5,464+ 7,416+7,448+ 11,338+)? (7,448+ is reserved by Dodson for 6 years with no result)

https://escatter11.fullerton.edu/nfs/numbers.php

7,395- is not P113*P201, it is P107*P114

Fixed. Thanks for letting me know!

Will the next numbers after 10,365+ be the quintic numbers 12,319- and 11,341- ?

Wait and see. A little patience does not hurt.