April 17, 1987


Page 48 is enclosed. It is taking longer now for pages to fill. Recall
that Page 47 was mailed last November. Some of you thought I had
stopped sending pages to you. But I have not stopped.

The first entry on Page 48 actually was sent with Page 47 as a last
minute bulletin. 2,634L c95 was on the "More Wanted List" mailed with
Page 47. When Silverman factored it, 2,674M c87 replaced it on the
list. The other "Wanted" numbers which were done on Page 48 are 3,181+
c82, 5,146+ c83, 6,121+ c83 and 7,104+ c82. te Riele did 7,104+ and
Silverman did the others. No "Most Wanted" numbers were done on this
page. Update # 5 will have new "Wanted" lists.

The first holes which were factored on Page 48 were the same numbers as
the "Wanted" numbers mentioned above. The factorizations numbered 2244,
2290 and 2295 were of the second holes in their tables. Those numbered
2247, 2248, 2268, 2275, 2283, 2289, 2292, 2293 and 2296 were of third
holes. Several fifth holes were done, but for some reason no fourth
holes happened to be factored on Page 48. Silverman did all of the
second, third and fifth holes on this page.

Silverman factored the last c75's on Page 48. Furthermore, now there
are only three c76's and two c77's left in Appendix C. I expect them to
be finished on Page 49.

Brent found the first known factor of the Mersenne number 2^631-1.

There were no new champions on Page 48. If we had a category for
Quadratic Sieve on a Supercomputer, then te Riele would be the new
champion for factoring 7,104+ c82 with a Cyber 205.  My factorization
of 6,170+ c77 on a VAX 8600 was the largest mp-qs factorization done by
a single, ordinary computer.

The smallest prime factor discovered on Page 48 has 16 digits.

Gary Gostin sent me six more divisors k.2^n + 1 of Fermat numbers F_m.
They are

     k     n     m
76432329   74    72 
   77377  550   547 
  733251  377   375 
  810373  378   376 
     501 3508  3506 
   13143 1454  1451 

Because of the slowness with which I am receiving new factors I expect
Update # 5 to appear before Page 49 is filled. Of course, I will send
you the partly filled Page 49 with Update # 5. A second edition of the
book is being prepared. It will contain the new factors through Update
# 5.  Updates after # 5 will give only the new factors found since
Update # 5 or, equivalently, those since the tables of the second
edition. Therefore, you should either save Update # 5 forever or buy
the second edition.

Here are the first two factors for Page 49:

2305 2,1330L c87 599169562712545924071281. c63 See next Silverman ecm
2306 2,1330L c63 19665145986649495362349061. p38 Silverman mp-qs

Keep the factors coming!


Sam Wagstaff