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


Chris Nash has probably found the following three primes asked. Here is his e-mail (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 indicates the next three digits of pi are 842... they are in fact 846.

The concatenation I have found is

<and the new terms>
846264338327........303906979207 (3057 digits!)

As explained above, the 3057-digit 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 primality-testing 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, 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 3057-digit 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 3057-digit probable prime with Primo. It took around 12 days. The Primo certificate was validated by Cert_Val and is available by e-mail request to

Chris correctly gives 73 and 467 as the next primes.

After this comes a 14650-digit 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 5020-digit prime and took 13 weeks to prove with Primo. The probable prime has been PRP-tested 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 pi-decimal 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.



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