Robert Harley

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...

INRIA, Domaine de Voluceau - Rocquencourt, 78153 Le Chesnay, France.

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