The Goldbach conjecture posits that any even number higher than 2 is the sum of two primes.
A googolplex
$G = 10^{10^{100}}$, a really humongous number with more zeroes than the atoms in the universe.
The current highest known Mersenne prime is
$2^{74207281}-1$, with 22338618 digits.
There isn't a known way of expressing a googolplex as the sum of two primes, and there won't be for the foreseeable future. It's just too difficult to prove that a large number is prime. However, it's often rather simple to prove that a number is not prime. The site alpertron has a lot of info about the divisors of numbers near a googolplex.
Is there something like a prime that we can aim for? Consider the numbers 24287 and 24289. It's easy to check that they aren't divisible by any numbers under 100. You might think this is a pair of twin primes, since they differ by 2. In actuality, neither number is prime,
$24287=149\times163$ and
$24289=107\times227$. The first number is 149-rough, and the second is 107-rough. A $k$-rough number has no prime factors under
$k$. Together they are a 107-rough pair since the smallest prime factor of their product is 107.
Divide a googolplex into two 100-rough numbers.
Solution:
$(\frac{G}{2} -3, \frac{G}{2} +3)$ are two numbers that add to
$G$ and have a product with smallest prime divisor 149.
We can sieve the numbers around
$G/2$ for the first hundred primes with the following code:
store = Table[0, {5000}];
Do[
p = Prime[k];
z = Mod[5 PowerMod[10, 10^100 - 1, Prime[k]], Prime[k]];
max = Min[{Floor[Abs[(2000 - z)]/p], Floor[Abs[(2000 - (p - z))]/p]}];
(*minus*) Do[
If[store[[z + p n]] == 0, store[[z + p n]] = p ], {n, 0, max}];
(*plus*) Do[
If[store[[(p - z) + p n]] == 0,
store[[(p - z) + p n]] = p ], {n, 0, max}];
,
{k, 1, 100}]
{231, 279, 321, 351, 381, 441, 477, 543, 549, 561, 573, 729, 783, 903} are positions of zeroes remaining in the table.
{677, 20063, 641, 4649, 9967, 5651, 63727, 1811, 557, 4793, 647, 2633, 0, 811} are the first prime divisors for those positions.
$(\frac{G}{2} -783, \frac{G}{2} +783)$ has no known divisors. Checked to 5852719141, so far.
{783, 2211, 2499, 3507, 4689, 6069, 6483, 6819, 7293, 7533, 7983, 8121, 8379, 8529, 8841, 9129} are the positions under 10000 with no known divisors so far.
I'd say a 236887691-rough split of a googolplex is pretty good, but I'll let this run for a few days to see how far I can push it.
UPDATE -- Here's the sieve after an overnight run. It's worth noting that my technique above is basically the Sieve of Eratosthenes. The first 273000000 primes have been checked, up to 5852719141. The list below is the status of hold-outs under 30000 last night. A "0" means the number is still a hold-out.
{{783, 0}, {2211, 1698813629}, {2499, 3624475301}, {3507, 275362319}, {4689, 0}, {6069, 0}, {6483, 0},
{6819, 669581027}, {7293, 0}, {7533, 601336979}, {7983, 0}, {8121, 0}, {8379, 0}, {8529, 0}, {8841, 0},
{9129, 0}, {12291, 0}, {12597, 3981247817}, {13737, 0}, {13857, 398751709}, {14013, 2133021211},
{14133, 0}, {14661, 0}, {15399, 0}, {15837, 0}, {16587, 368517221}, {16689, 1665315719}, {17343, 0},
{17721, 0}, {17811, 0}, {20121, 263680121}, {20313, 0}, {20583, 0}, {20709, 0}, {21141, 0}, {21249, 0},
{21321, 0}, {21729, 0}, {21771, 0}, {22779, 0}, {23013, 0}, {24423, 258700111}, {25707, 0}, {26799, 0},
{27663, 0}, {28641, 0}, {28797, 269471659}}
$(a = \frac{G}{2} -2499, b = \frac{G}{2} +2499), a+b$ = googolplex.
$a b$ has smallest prime factor 3624475301
What are the odds that one of (G/2)±783 is actually prime? By the prime number theorem, a random number near a googolplex has a 1 in (googol log(10)) chance of being prime. But these are sieved random numbers.
$\Phi(x, x^{1/u}) = e^{-\gamma} \frac{x}{log(x^{1/u})}$ -- The Buchstab function for counting rough numbers.
What are the odds that a random 10^(10)-rough number near a googol is prime?
log(10)/ (10^(98)) -- odds random number near googol is prime.
e^? log(10) / (10^(99)) -- odds random number near googol is 10^(10)-rough.
e^?/10 ~ 0.178107 -- odds random 10^(10)-rough number near a googol is prime.
Too early in the morning for me to calculate the odds near a googolplex.