MPhil thesis - on k-tuples of almost primes

We conjecture that any "good" set of k linear functions:
L1(n)=a1*n+b1,  ...  , Lk(n)=ak*n+bk,
can be prime numbers simultaneously for infinitely many n.

e.g.
k=2, {n, 2n+1}: {3, 7}, {5, 11}, ...
k=3, {n, n+2, n+6}: {5, 7, 11}, {17, 19, 23}, ...
k=4, {10n+1, 10n+3, 10n+7, 10n+9}: {11, 13, 17, 19}, {101, 103, 107, 109}, ...

This is called the prime k-tuple problem.

The case k=2, {n, n+2}, corresponds to the twin prime problem.
The case k=4, {n, n+2, n+6, n+8}, corresponds to the prime decade problem.

It is also conjectured that infinitely many often, {n, n+2} can be simultaneously E2-numbers, i.e. integers which are products of exactly two primes.
e.g. {3*11, 5*7}, {7*13, 3*31}, {19*41, 11*71}, ...
This can be called the twin E2 conjecture.

Results achieved:

This project shows that infinitely many often,

I. n is an E2, at the same time n+2 is either an E2 or a prime.

II. n is a prime, at the same time n+2 has at most 3 prime factors.

III. n has at most 5 prime factors, at the same time n+2 has at most 4, n+6 has at most 3, and n+8 has at most 2 prime factors.

These results can apply to any "good" 2 and 4-tuples too.

Finally, let 「k: rk, Rk」 denote the statement:

"For any 'good' k-tuple, infinitely many often the k tuples altogether have at most rk prime factors, at the same time each individual tuple has at most Rk prime factors."

Then IV. 「3: 8, 3」,「4: 12, 4」「4: 11, 5」and「5: 15, 5」are proved as well.