Several sets of numbers have been factorised recently. These include the following:

1) N = k • 2ⁿ ± 1 where k is 3, 5, 7, 9, 11, 13 or 15 and n is up to 1000.  The results are recorded on a web-site by Mikael Klasson who has written some PHP scripts for direct submission of factors.

2) Numbers with repeated digits like 2 • 10¹⁵⁷ - 1 or the digit 1 followed by 157 nines, written as 19₁₅₇.  See the web-site of Makoto Kamada for a large database of these numbers.  This site has the added advantage of allowing secure number reservation as well as submission of results.

3) N = x^y + y^x for 1 < y < x < 151.  First proposed by Paul Leyland, the results are now collected on the web-site of Andrey Kulsha.

4) (NEW) N(x,y) = x^y • y^x + 1 with 1 < y < x.  I have been looking at these numbers myself recently.  They are a variant of type (3) and have the advantage of quickly producing very large and interesting numbers.

For example, N(100, 99) is a 398 digit number: 3660323412732 ... 0000000000001. The disadvantage is that they also produce very small numbers.  N(x,1) is just x + 1 (not very interesting).  N(x,2) = x² • 2^x + 1 slightly more interesting and so on.  Tables will be published here Real Soon Now.

These N(x,y) numbers also produce large primes.  One from my tables is the 109 digit number:

All of these numbers are amenable to attack using the Number Field Sieve method (even the third group, above, using large modulo fractions in the data input), and so can be factorised using GGNFS.