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Fix a base b and partition the nonzero residues modulo a prime p>b by the leading digit of r/p. A multiplier is globally silent when it moves every residue out of its original digit bin. A collision coordinate separates the silent multipliers from the constructive ones. At a fixed lag it gives an exact finite recipe for the globally silent primes. In base ten, the primes 11,13,19,23,37,41, and 73 are exactly the primes greater than ten whose repetends are silent at lag one.
For p>b+1, the constructive side gives a signed sawtooth formula for the collision count, its exact mean, and an exact interval-overlap formula for its variance. The diagonal part of the variance is \frac23Q+O_b(1), where Q=\lfloor(p-1)/b\rfloor. Every unresolved contribution is isolated in a finite off-diagonal overlap ledger. For each fixed base b\geq3, finite calculations suggest that the full variance divided by Q tends to 1-1/b. The variance-scale conjecture is reduced to the signed off-diagonal overlap ledger.
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}\}}.
Definition 1. A prime is repetend-silent at lag \ell when R_{p,b}(\ell)=0. It is globally silent at lag \ell when C_{p,b}([b^\ell]_p)=0.
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 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.
Lemma 2 (Conjugation). The map r\mapsto[br]_p permutes the nonzero residues and satisfies \delta_{p,b}(r)=\delta_{p,b}(s) \quad\Longleftrightarrow\quad [br]_p\equiv[bs]_p\pmod b. Consequently, C_{p,b}(g) =\#\{x\in\{1,\ldots,p-1\}\mid x\equiv[gx]_p\pmod b\}.
Proof. Euclidean division gives br=p\delta_{p,b}(r)+[br]_p. Reduction modulo b gives [br]_p\equiv-p\delta_{p,b}(r)\pmod b. The prime p>b is invertible modulo b, and the digit values form a complete set of representatives modulo b. This proves the first claim. Apply it to r and [gr]_p, then replace [br]_p by x. ◻
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.
Theorem 3 (Collision coordinate). For g\neq1 and c=c(g), C_{p,b}(g) =2\#\left\{m\in\{1,\ldots,Q\}\ \middle|\ [mc]_p>mb\right\}.
Proof. By Lemma 2, a collision is a residue x for which [gx]_p=x+mb for some integer m. The multiplier is not the identity, so m is nonzero. Both residues lie between 1 and p-1, which gives 1\leq |m|\leq Q.
The relation c(1-g)\equiv b\pmod p turns the collision equation into x+mc\equiv0\pmod p. For m>0, the only possible residue is x=p-[mc]_p. The condition [gx]_p=x+mb<p holds exactly when [mc]_p>mb. For m<0, write m=-n. The only possible residue is x=[nc]_p, and [gx]_p=x-nb>0 holds under the same condition [nc]_p>nb. Each successful positive index therefore gives two collisions, one for each sign. No other collisions occur. ◻
Corollary 4 (The zero gate). The collision count vanishes exactly when c(g)\in\{1,\ldots,b-1\}. Equivalently, the bin-deranging multipliers are g\equiv-\frac{u}{b-u}\pmod p, \qquad 1\leq u<b. There are exactly b-1 of them. Every collision count is even.
Proof. If c<b, then mc<p and [mc]_p=mc<mb for every m\leq Q. If c>b, the index m=1 satisfies [c]_p=c>b. Theorem 3 proves the zero criterion and evenness. Solving c(1-g)=b and putting u=b-c gives (10). Distinct values of u give distinct multipliers because p>b. ◻
The zero gate turns silence at a fixed lag into a finite factorization problem.
Theorem 5 (Global silent-prime recipe). Fix b\geq2 and \ell\geq1. A prime p>b is globally silent at lag \ell if and only if it divides N_{b,\ell}(u) =b^{\ell+1}-u(b^\ell-1) for some u\in\{1,\ldots,b-1\}. The b-1 integers in (11) are positive and form an arithmetic progression with first term b^\ell(b-1)+1 and common difference -(b^\ell-1).
Proof. The identity multiplier is not silent. For every other multiplier, Corollary 4 gives [b^\ell]_p\equiv-\frac{u}{b-u}\pmod p for some u in the stated range. The denominator is nonzero modulo p. Clearing it gives b^\ell(b-u)+u\equiv0\pmod p, which is precisely p\mid N_{b,\ell}(u). Every step is reversible. Indeed, the divisor condition cannot produce the identity multiplier. If b^\ell\equiv1\pmod p, then b^\ell(b-u)+u\equiv b\not\equiv0\pmod p because p>b.
The smallest value occurs at u=b-1 and equals b^\ell+b-1, so all terms are positive. The progression follows directly from N_{b,\ell}(u)=b^{\ell+1}-u(b^\ell-1). ◻
Corollary 6. Only finitely many globally silent primes occur at any fixed base and lag.
The same finiteness holds for the weaker repetend notion, although its complete list may be larger.
Theorem 7 (Repetend bound). If p is repetend-silent at lag \ell, then p\leq b^{\ell+1}. Hence only finitely many repetend-silent primes occur at a fixed base and lag.
Proof. Suppose p>b^{\ell+1}. The orbit positions represented by 1 and b^\ell both have digit zero because \delta_{p,b}(1)=0, \qquad \delta_{p,b}(b^\ell) =\left\lfloor\frac{b^{\ell+1}}p\right\rfloor=0. They give a match in (4). The repetend is therefore not silent. ◻
For base ten and lag one, the global recipe gives the nine integers in Table 1.
| u | 100-9u | factorization |
|---|---|---|
| 1 | 91 | 7\cdot13 |
| 2 | 82 | 2\cdot41 |
| 3 | 73 | prime |
| 4 | 64 | 2^6 |
| 5 | 55 | 5\cdot11 |
| 6 | 46 | 2\cdot23 |
| 7 | 37 | prime |
| 8 | 28 | 2^2\cdot7 |
| 9 | 19 | prime |
Corollary 8 (The seven lag-one primes). In base ten, the repetend-silent primes greater than ten at lag one are exactly \{11,13,19,23,37,41,73\}.
Proof. Every prime in (13) is globally silent by Theorem 5 and Table 1, so it is repetend-silent. Theorem 7 rules out every prime greater than 100.
It remains to inspect the primes between 10 and 100 that do not occur in (13). Table 2 gives an exponent j and one pair (r,[10r]_p) with r=[10^j]_p for each such prime. The last entry in the pair column is their common digit. Every remaining prime therefore has a lag-one match. ◻
| p | j | (r,[10r]_p;\,d) | p | j | (r,[10r]_p;\,d) |
|---|---|---|---|---|---|
| 17 | 2 | (15,14;\,8) | 61 | 6 | (27,26;\,4) |
| 29 | 2 | (13,14;\,4) | 67 | 22 | (37,35;\,5) |
| 31 | 2 | (7,8;\,2) | 71 | 11 | (16,18;\,2) |
| 43 | 4 | (24,25;\,5) | 79 | 6 | (18,22;\,2) |
| 47 | 10 | (21,22;\,4) | 83 | 9 | (64,59;\,7) |
| 53 | 2 | (47,46;\,8) | 89 | 1 | (10,11;\,1) |
| 59 | 15 | (52,48;\,8) | 97 | 24 | (75,71;\,7) |
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.
For a nonintegral real number x, write ((x))=\{x\}-\frac12. None of the arguments below is integral.
Corollary 9 (Sawtooth form). Let g\neq1 be constructive and let c=c(g)>b. Then C_{p,b}(g) =Q+\sum_{m=1}^{Q} \operatorname{sgn}\left( ((mc/p))-((mb/p)) \right).
Proof. The sign in (14) is positive exactly when [mc]_p>mb. Equality would imply m(c-b)\equiv0\pmod p, which is impossible because m<p and c\neq b. Replace the indicator in (9) by one half of one plus the sign. ◻
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.
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}.
Theorem 10 (Exact mean). With p-1=bQ+R as in (6), \mathbb E C =\frac{Q\bigl(b(Q-1)+2R\bigr)}{N} =Q+\frac{QR}{N}.
Proof. Fix m\leq Q. Multiplication by m permutes the nonzero residues modulo p. Exactly p-1-bm of them are greater than bm. The omitted parameters c=1,\ldots,b never contribute because mc\leq mb<p. Therefore \#\{c\in\mathcal A_{p,b}\mid[mc]_p>mb\}=p-1-bm. Sum this identity over m and apply Theorem 3. Substitution of p-1=bQ+R gives (16). ◻
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).
Theorem 11 (Exact interval-overlap variance). The quantities in (18) satisfy H_{mn}=|\mathcal I_m\cap\mathcal I_n|. The collision variance is exactly V_{p,b}=\operatorname{Var}(C) =4\left\{ \frac1N\sum_{m,n=1}^{Q}H_{mn} -\left(\frac{bQ(Q-1)}{2N}\right)^2 \right\}.
Proof. For c\in\mathcal A_{p,b}, Y_m(c) =\left\lfloor\frac{mc}{p}\right\rfloor -\left\lfloor\frac{m(c-b)}p\right\rfloor. The two arguments differ by mb/p<1, so the difference is either zero or one. It equals one exactly when the interval (m(c-b),mc] contains a multiple kp. Solving for the integer c gives the union in (17). Consecutive starting points are separated by at least b, since p/m>b for m\leq Q. The first starting point is greater than b. The last endpoint is p-\left\lfloor\frac pm\right\rfloor+b-1\leq p-1. Thus every interval lies in \mathcal A_{p,b}. The intervals are disjoint and |\mathcal I_m|=b(m-1).
Two integer intervals of length b with starting points a and d intersect in (b-|a-d|)_+ points. Disjointness within each family shows that (18) counts every point of \mathcal I_m\cap\mathcal I_n exactly once.
The collision-coordinate theorem gives C=2Q-2\sum_mY_m. Hence \operatorname{Var}(C)=4\operatorname{Var}\left(\sum_mY_m\right). The first term in the second moment is \mathbb E\left(\sum_mY_m\right)^2 =\frac1N\sum_{m,n}H_{mn}. Equation (21) gives \mathbb E\sum_mY_m =\frac1N\sum_{m=1}^{Q}b(m-1) =\frac{bQ(Q-1)}{2N}. Subtracting the square of the mean proves (19). ◻
The exact formula separates into a diagonal part and an off-diagonal part.
Corollary 12 (Diagonal contribution). Write V_{p,b}=V_{p,b}^{\mathrm{diag}}+V_{p,b}^{\mathrm{off}}. The two terms are V_{p,b}^{\mathrm{diag}} =4\left( \frac{bQ(Q-1)}{2N} -\frac{b^2Q(Q-1)(2Q-1)}{6N^2} \right) and V_{p,b}^{\mathrm{off}} =8\sum_{1\leq m<n\leq Q} \left( \frac{H_{mn}}N -\frac{b^2(m-1)(n-1)}{N^2} \right). For each fixed base, V_{p,b}^{\mathrm{diag}}=\frac23Q+O_b(1).
Proof. The diagonal term is four times the sum of the individual variances of the indicators Y_m. Use \mathbb EY_m=b(m-1)/N and the standard formulas for the sums of the first Q-1 integers and their squares. The off-diagonal term is eight times the sum of their pairwise covariances. Finally, N=bQ+O_b(1) in (22), which gives (24). ◻
The diagonal law is proved. The leading behavior of the full variance is the signed off-diagonal overlap sum in (23).
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].
| p | Q | V_{p,10} | V_{p,10}/Q |
|---|---|---|---|
| 97 | 9 | 6.00108 | 0.66679 |
| 193 | 19 | 13.96740 | 0.73513 |
| 499 | 49 | 40.45310 | 0.82557 |
| 997 | 99 | 84.35310 | 0.85205 |
| 1999 | 199 | 173.09915 | 0.86984 |
| 4999 | 499 | 440.54874 | 0.88286 |
| 9973 | 997 | 884.82743 | 0.88749 |
At the common prime 9973, the same calculation gives the values in Table 4. They are close to 1-1/b from below.
| b | Q | V_{9973,b}/Q | 1-1/b |
|---|---|---|---|
| 3 | 3324 | 0.666072 | 0.666667 |
| 6 | 1662 | 0.828314 | 0.833333 |
| 7 | 1424 | 0.847494 | 0.857143 |
| 10 | 997 | 0.887490 | 0.900000 |
| 12 | 831 | 0.905998 | 0.916667 |
The displayed values lead to the following conjecture.
Conjecture 13 (Variance scale). For each fixed base b\geq3, \frac{V_{p,b}}{Q}\longrightarrow1-\frac1b as p tends to infinity through the primes.
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 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.
[1]A. S. Petty, Bin Derangements and the Gate Width Theorem, July 2022. \href{https://doi.org/10.5281/zenodo.21850917} {doi:10.5281/zenodo.21850917}.
[2]A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inform. Theory 20 (1974), no. 1, 90–94. https://doi.org/10.1109/TIT.1974.1055169.
[3]S. C. Kak and A. Chatterjee, On decimal sequences, IEEE Trans. Inform. Theory 27 (1981), no. 5, 647–652. https://doi.org/10.1109/TIT.1981.1056394.
[4]N. J. Armstrong and R. J. Armstrong, Some properties of repetends, Math. Gaz. 87 (2003), no. 510, 437–443. https://doi.org/10.1017/S0025557200173619.
[5]K. Girstmair, Digit variance and Dedekind sums, J. Number Theory 65 (1997), no. 2, 197–205. https://doi.org/10.1006/jnth.1997.2149.
[6]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
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