Problems & Puzzles:
Problems
Problem 18. Pi as
a concatenation of the smallest contiguous different
primes
…
has the following smallest contiguous different primes: 3, 14159,
2, 653,
5, 89,
7, 9323
Find the following 3 primes
Solution
Chris Nash has probably
found the following three primes asked. Here is his
email (6/10/99)
"I 'probably' know the next 3 primes in the
sequence detailed in Problem 18  the representation of
pi as the concatenation of the smallest contiguous
different primes. The answer is "probably",
because the next term in the sequence is at the moment
only known to be a probable prime (it has now been tested
to a very large number of bases). By the way, the graphic
on the page http://www.sci.net.mx/~crivera/problems/prob_018.htm
indicates the next three digits of pi are 842... they are
in fact 846.
The concatenation I have found is
3
14159
2
653
5
89
7
9323
<and the new terms>
846264338327........303906979207
(3057 digits!)
73
467
As explained above, the 3057digit term is currently only
known (Euler) probable prime, and requires some
considerable work on factoring to prove its
primality. I hope for a lucky factor to show soon
in order to complete the proof. However it has been
proven and verified that all shorter digit sequences up
to 3056 digits are certainly composite.
The primality testing was done with the PrimeForm
classical primalitytesting program which I am currently
developing, along with a sieving phase which removed
several possibilities from consideration before
attempting the primality tests. More details about
PrimeForm can be found at
http://welcome.to/primeform,
and I forward this message to the PrimeForm user group.
Of course, the problem remains  how much further can
this sequence continue, and can the 3057digit term be
proven prime?"
***
A new visitor to an old problem!
The 14/9/2002 Jens
Kruse Andersen, confirmed the primality of the 3057 digits Nash's
prime using PRIMO by 12 days. He also made this:
a)
confirmed the two next primes devised by Nash
b) got the next probable prime (14650 digits)
c) got the next four small primes and
d) points out that the next prime must have more than 32000 digits!
Here is his email:
I have used a C program with the Miracle big
integer library and Primeform/GW. My computation of the next three
primes agrees with Chris Nash. I have proved the 3057digit
probable prime with Primo. It took around 12 days. The Primo
certificate was validated by Cert_Val and is available by email request
to jens.k.a@get2net.dk
Chris correctly gives 73 and 467 as the next
primes.
After this comes a 14650digit probable prime 
beyond the range of current programs and computers to prove, unless a very
lucky factorization of n+1 is found. For comparison, the "general" prime
record is currently (Sep 2002) a 5020digit prime and took 13 weeks to
prove with Primo. The probable prime has been PRPtested in many
bases and is practically guaranteed to be prime.
If the prime sequence is extended further then there
will probably soon be a case where the first pidecimal after a prime is
0. This must be considered a leading 0 to the next prime if the sequence
shall continue.
This is how far I got:
3, 14159, 2, 653, 5, 89, 7, 9323,
8462643383.....3906979207 (3057 digits, proven
prime),
73, 467,
2218256259.....5939500279 (14650 digits, probable
prime),
3389, 59,
2057668278967764453184040418554010435134838953, 1201,
3263783692.....?????????? (at least 32000 digits)
I expect the sequence of primes to be infinite.
The density of primes gives this expectation for all
infinite digit lists where each digit is "random"  pi is not random but I
guess the decimals are close enough for this purpose.
***
