Long division and silence

Long division can fall completely silent. In base ten, \frac1{13}=0.\overline{076923}. A one-place cyclic shift gives 769230. None of its six digits agrees with the digit beneath it. The neighboring prime behaves differently. The repeating block of 1/17 leaves two digits fixed under the same shift. Periodicity alone does not explain the zero at 13.

Silence is the zero edge of the collision count. Once that edge is exact, the same coordinate continues through the constructive multipliers and exposes their variance. The question is therefore not only which primes are silent, but how the remaining collision counts spread when silence breaks.

There are two natural ways to ask for silence. One may inspect only the repetend of 1/p, or one may ask the complete digit partition to be silent. They agree when the powers of the base run through every nonzero residue. They need not agree on a shorter orbit. Keeping the two notions separate is essential.

Fix an integer base b\geq2 and a prime p>b. Let [a]_p denote the least positive representative of a nonzero residue modulo p. Define \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor, \qquad 1\leq r<p. The values 0,\ldots,b-1 are the leading base-b digits of the fractions r/p.

For g\in\mathbb F_p^\times, define the collision count C_{p,b}(g) =\#\left\{r\in\{1,\ldots,p-1\}\ \middle|\ \delta_{p,b}(r)=\delta_{p,b}([gr]_p)\right\}. A nonidentity multiplier is bin deranging when its collision count is zero and constructive when its collision count is positive.

Let L_p=\operatorname{ord}_p(b) and put d_j=\delta_{p,b}([b^j]_p), \qquad j\in\mathbb Z/L_p\mathbb Z. This is the repeating digit word of 1/p. Its lag-\ell autocorrelation count is R_{p,b}(\ell) =\sum_{j\bmod L_p}\mathbf1_{\{d_j=d_{j+\ell}\}}.

Global silence implies repetend silence because the powers of b form one multiplicative orbit. If b is a primitive root modulo p, that orbit is all of \mathbb F_p^\times and R_{p,b}(\ell)=C_{p,b}([b^\ell]_p).

Coordinatewise agreement under a cyclic shift is Hamming correlation. Lempel and Greenberger place this statistic in the general theory of periodic-sequence correlation [2]. Kak and Chatterjee study Hamming distance between prime-reciprocal digit sequences and their cyclic shifts, together with numerical autocorrelation bounds and a condition for zero cross-correlation [3]. Repetend multiplication and its modular orbit structure are classical [4].

Digit-value variance follows a separate arithmetic line. Girstmair expresses the variance of a full-period reciprocal digit word in terms of a Dedekind sum and studies its fixed-base behavior as the prime grows [5]. The variance considered here is not a variance of digit values along one orbit. It is the population variance, over constructive multipliers at one prime, of the complete-group equality count C_{p,b}(g). Equation (5) links this collision count to orbit Hamming correlation when the base is primitive; the full-group definition also covers shorter orbits. The finite silent-prime recipe and the interval-overlap variance formula preserve that distinction. The full variance asymptotic remains conjectural.

The collision coordinate

The interval bins in (1) become ordinary residue classes after one permutation. This conjugation supplies both the zero gate and the coordinate used for the variance.

Bin Derangements and the Gate Width Theorem [1] gives the rational family of zero multipliers. The collision coordinate also counts the positive comparisons, which determine the constructive mean and variance.

Write p-1=bQ+R, \qquad 0\leq R<b. For a nonidentity multiplier g, define its collision coordinate by c(g)=\left[b(1-g)^{-1}\right]_p. The value b cannot occur because it would force g=0. The map g\mapsto c(g) is a bijection from \mathbb F_p^\times\setminus\{1\} onto \{1,\ldots,p-1\}\setminus\{b\}, with inverse g_c=1-bc^{-1}\pmod p.

A finite recipe

The zero gate turns silence at a fixed lag into a finite factorization problem.

The same finiteness holds for the weaker repetend notion, although its complete list may be larger.

For base ten and lag one, the global recipe gives the nine integers in Table 1.

The distinction between the two notions of silence cannot be removed in general. In base eight, the prime 13 has orbit 1,8,12,5 and digit word 0,4,7,3, which is repetend-silent at lag one. It is not globally silent because r=2 gives \delta_{13,8}(2)=\delta_{13,8}(3)=1, \qquad [8\cdot2]_{13}=3.

Signed collision comparisons

For a nonintegral real number x, write ((x))=\{x\}-\frac12. None of the arguments below is integral.

This is a deterministic identity. It does not assert that the signs are independent. Their dependence is exactly the issue measured by the variance ledger below.

The variance ledger

Assume p>b+1 and put \mathcal A_{p,b}=\{b+1,\ldots,p-1\}, \qquad N=|\mathcal A_{p,b}|=p-b-1. Through (8), this is exactly the set of constructive nonidentity multipliers. Write C(c)=C_{p,b}(g_c), \qquad c\in\mathcal A_{p,b}. Let \mathbb E and \operatorname{Var} denote the population mean and variance over the uniform set \mathcal A_{p,b}.

The second moment has an exact geometric form. Define X_m(c)=\mathbf1_{\{[mc]_p>mb\}}, \qquad Y_m(c)=1-X_m(c). For m\geq2, let \mathcal I_m =\bigcup_{k=1}^{m-1} \left\{ \left\lceil\frac{kp}{m}\right\rceil, \ldots, \left\lceil\frac{kp}{m}\right\rceil+b-1 \right\}, and put \mathcal I_1=\varnothing. The intervals in each union are disjoint. For 1\leq m,n\leq Q, define H_{mn} =\sum_{k=1}^{m-1}\sum_{\ell=1}^{n-1} \left( b- \left| \left\lceil\frac{kp}{m}\right\rceil -\left\lceil\frac{\ell p}{n}\right\rceil \right| \right)_+, where (x)_+=\max(x,0).

The exact formula separates into a diagonal part and an off-diagonal part.

The diagonal law is proved. The leading behavior of the full variance is the signed off-diagonal overlap sum in (23).

The variance scale at finite primes

Table 3 records exact population variances computed from the definition. The displayed decimals are rounded from those exact values.

The finite calculations in this section were performed with nfield [6].

At the common prime 9973, the same calculation gives the values in Table 4. They are close to 1-1/b from below.

The displayed values lead to the following conjecture.

The conjecture concerns the leading term. The remainder is open. Corollary 12 proves the diagonal contribution and reduces the conjecture to the signed overlap sum in (23).

The open overlap

The finite zero set is exact, the diagonal law is proved, and the full variance scale has been reduced to one concrete object. For every prime, each term of (23) is an explicit intersection of two interval families. The remaining problem is to determine the leading cancellation in that signed overlap ledger as the prime varies.

References