Hi all,

Anyone help me on this?

Let P(n) be the smallest string of 1's and 0's that is prime in every
base from 2 to n. Let Pb(n) be the decimal version of P(n) expanded in
base b.

Example: P(5)=10010111 since P2(5)=151, P3(5)=2281, P4(5)=16661 and
P5(5)=78781 are all prime, and 10010111 is the smallest such string.

I've done a little work on this and have found P(n) up to P(9) (=
10011110011011110110110011). P(10) has at least 29 digits and the
number of 1's in it must be prime to 210, but I haven't managed to find
it yet.

Questions:

1. What is P(10)?

2. Is there any theoretical reason that P(n) should exist for all n? Or
maybe a proof that they don't?

3. Can anyone establish any bounds for the values of P(n)?

Many thanks

Richard