- Elsewhere (sci.crypt/sci.math "Base-b digits of primes and their reciprocals"),
the correctness of part of a paper by Kak in Cryptologia, Jan 1989, has been
called into question.
The D-sequence, in radix r, for 1/q, q prime, is written as
a_1 a_2 ... a_k, where 1/q = a_1/r^1 + ... + a_k/r^k + ...
and a_i = a_(k+i) for all i > 0.
Let q = b_n r^n + ... + b_1 r + b_0; then n will be called
the _degree_ of q in radix r.
Theorem 1. Given that the prime generating a D-sequence has
degree n or less, any 2n+2 digits suffice to find the prime.
But Kak's "Theorem 1" cannot be correct, as shown by the
1/17 = 0.(0588235294117647)
1/19 = 0.(052631578947368421)
If the theorem were correct, only *one* 2-digit prime could
have the 4-digit subsequence 05xxxx5xxxxxx6 in the D-sequence
for its reciprocal -- but we have here two such primes.
Other base-10 counterexamples are abundant; for example,
the D-sequence for 1/109 matches that of 1/113 in 10 places
and it matches that for 1/149 in 12 places.
>>>Alas I couldn't find an online copy of the paper, as I think it could be an
interesting exercise to find the bullet which can put Kak's non-proof out of
its misery. I suspect that there may be other like-minded individuals here on
If anyone does find it (the paper, or the bullet), please do get back.
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