--- In

primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> Another hint: a quotient too big for the margin.

Kevin has asked me to post my solution.

The key to his puzzle was

http://en.wikipedia.org/wiki/Fermat_quotient
With base a = 1993, the Fermat quotient

(a^(p-1) - 1)/p is divisible by p for

p = 2, 5, 257, 1399, 1667, 2011

and for no other prime p < 10^10. In other words,

2011 is the largest known Wieferich prime in base 1993.

Another puzzle is to find a sequence of increasing primes

such that p[n+1] is the smallest Wieferich prime in base p[n]

for which p[n+1] > p[n].

Let's call this a "generalized Wieferich sequence".

For example, the primes

1153, 1747, 1993, 2011, 32933, 16365127

form a generalized Wieferich sequence of length 6, with

16365127 being Kevin's correct solution to my

"1993/2011/32933 puzzle".

Puzzle 7: Find a generalized Wieferich sequence of length > 6.

Comments: This may be solved using primes less than 10^9. So far,

I have not found a generalized Wieferich sequence of length > 7.

David