*The previous arithmetic progression record was discovered by ***Arlin
Anderso**n, using the following method. Suppose two primes k1.2^n+1 and
k2.2^n+1 are known. Write k=(k1+k2)/2, then k.2^n+1 would complete an
arithmetic progression and may be tested for primality. After trying
enough pairs, **Arlin** was successful, but his value of k was even,
and unknown to him, the third entry was already in the database in the
form with k odd and a different power of n. (Nevertheless, he is credited
at http://www.utm.edu/research/primes/lists/top20/ArithmeticProg2.html
with the discovery that the three primes were in arithmetic progression).

**CN** suggested the following method. Take two large sets of primes
X and Y. Then for every pair, one prime x from X, and one prime y from Y,
test (x+y)/2. This not only provides a lot of possibilities, but if the
sets X and Y are sets of ** Proth** primes k.2^n+1, even with different values
of n, such primes are provable by classical methods. It is only necessary
for the smallest value of n to be a little more than 1/3 the larger.

**CR** provided the 347 primes in set X, and **CN** chose the set
Y to be those Proth primes with over 20,000 digits. For each member of Y, **CN**
generated a file with 347 numbers to test. **CR** used **PrimeForm/GW**
to test the numbers for probable primality, and **PrimeForm** for
Windows to prove the result. The new record is

*largest:*

**
24094785*2^67334+1** (20277 digits), previously discovered by **Michael
Hannigan**. (BTW, this - the largest - is the only prime of the triplet that
is shown in the **Caldwell's** page
corresponding to these kind of records, according to his rules)

*smallest:*

**
272238423*2^45454+1** (13692 digits), from **Carlos**' list.

*and the new prime as middle term:*

**
(24094785*2^21880+272238423)*2^45453 + 1 ** (20277 digits)

*Together they form an arithmetic progression with common difference
(24094785*2^21880-272238423)*2^45453 + 1*

*Questions:*

*1) Using this method, can you find (and
prove!) a new, higher, arithmetic progression record? It seems since our
search did not take very long, this should not be difficult.*

*2) Is there a better method, that will produce
larger A.P.'s, or discover new A.P.'s more quickly? Of course, the records
discovered *must* be provable!*

*3) What about arithmetic progressions of
length greater than 3?*