**Hartley** wrote (Prime Curios!,
13298267):

Each of the 30 numbers 13298267 + 1887270 k, k=0..29
is a "semiprime" (a product of two
primes). There is no longer
Arithmetic Progression of semiprimes whose terms are all less than 10^8

**Question:**

**1. Do you devise a systematic
approach in order to get larger progressions?**

**2. F****ind
a larger progression, lets say 50, 100 and 200, semiprimes in A.P.**

**Solution:**

**J. K. Andersen** wrote this:

1. Computing all small semi primes is easy
compared to finding long arithmetic progressions among them. A
progression of at least 25 semi primes must have difference divisible by
5# = 30 to avoid factors 2^2, 3^2 and 5^2. To get a higher density
(around 43% semi primes) I used 11# = 2310 in the form c+2310x, where
c<2310 has no factor <=11. For each c, a byte array (faster than bit

operations) was set to 0 or 1 at position x to
indicate semi primes. A small x interval fitting in the cache memory was
tested at a time, without looking for progressions spanning more than
one interval. For every pair of semi primes with difference small enough
to potentially extend to 30 semi primes within an x interval, array
lookup was used to see how many semi primes were in arithmetic
progression.

2. The complexity grows exponentially and 50 semi
primes seems infeasible. 100 is completely out of the question, although
I guess arbitrarily long progressions exist. The best I found in a few
hours was 8 more cases with 30 and 2 with 31: 1852175623 + 6054510 k,
k=0..30 2522638201 + 445830 k, k=0..30 The search was not exhaustive.

***

Giovanni Resta wrote (March 19, 08):

I found one progression of length 35: 71243983 +
k*39029760 for k=0,...,34,

and some progressions of length 32, like 3900258487+k*103330920 and
25354741 + k*95795700, for k=0,...,31.

***