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).

The collision coordinate

The interval bins in (1) become ordinary residue classes after one permutation. This is the zero-gate geometry developed in Bin Derangements and the Gate Width Theorem [1], derived here in the form needed for silence 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

The finite calculations were performed with nfield [2]. Table 3 gives exact population variances computed from the definition. The displayed decimals are rounded. The nfield repository is the executable front door for inspecting the underlying silence, collision, and overlap objects across other finite parameters.

At the common prime 9973, the same nfield [2] 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