- The only known factor of the generalized Fermat number 10^1024+1:
- 1856104284667693057
- (found using some crumby ECM code).

- Some small prime factors of googolplex+1 = 10^(10^100)+1:
- 316912650057057350374175801344000001
- 155409907106060194289411023528840396801
- 1467607904432329964944690923937202176001
- 11438565962996913740067907829760000000001
- 495176015714152109959649689600000000000001
- 7399415202816574979127045311692800000000001
- 9823157208340761024963422324575436800000001
- 493333612765059415097397477376000000000000000000001
- 8019958276735747672058735099904000000000000000000001
- 8301034833169298227200000000000000000000000000000000000000001
- 325123864299130847232000000000000000000000000000000000000000001
- 35343349155678178508800000000000000000000000000000000000000000000000001
- 156941061512238345486336000000000000000000000000000000000000000000000001
- 370791604769783808000000000000000000000000000000000000000000000000000000000000001
- (found by sieving arithmetic progressions suggested by a theorem of Legendre).

One day during my lunch break I wrote a simple Eratosthenes' sieve to count twin primes. For instance, the number of pairs up to 10¹³ is 15834664872. The sum of inverses of twin primes up to that limit is about 1.81394376067 which extrapolates to an approximation for Brun's constant: 1.90216055 or 6 or 7 or thereabouts. Here is a 4.5K gif of a graph showing how the extrapolated sum converges as the limit is increased. Since then, Thomas R. Nicely has published the results of similar computations up to 10^14...

Talking about sums-of-inverses-of-primes, you may know that the sum of 1/p for primes p <= x is close to B + log log x. As far as I know, nobody has previously computed B to any significant precision. however B =

- 0.26149 72128 47642 78375 54268 38608 69585 90515 66648 26119
- 92061 92064 21392 49245 10897 36820 97141 42631 43424 66510
- 51617 72887 64860 21997 78339 03242 70044 42454 34874 01972
- 38640 66619 49557 09392 58171 27747 74211 98525 88072 66272
- 06414 44642 32590 02354 31051 77232 17392 56632 29980 31476 38316+

I gave a short talk about frequent gaps between primes at the 1994 Western Number Theory Conference in San Diego. Read all about it here!

All that stuff is like five years old.

Now here are some interesting numbers: 92221507219705345685350, 276856274258963891889538, 333373190151749761757285479, 41871609686648820507900581, 112815344736959642274131295791, 37837308472231540269443981458, 64206457470187038759216338447, 47455661896223045299748316018941. Check 'em out! Anyway, they're discrete logarithms in the groups of points on some elliptic curves over some finite fields and they are the answers to challenges posed by Certicom...

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