Doklady of the Academy of Sciences of the USSR
1962. Volume 143, No. 6

MATHEMATICS

E. V. NOVOSELOV

SOME FORMULAS CONNECTED WITH THE REDUCED SYSTEM OF RESIDUES

(Presented by Academician I. M. Vinogradov on December 8, 1961)

Introduce the following notation: \(m=\prod_p p^{k_p}\) is a fixed natural number; \(k_p\ge 0\) and is different from zero for only finitely many primes \(p\). \(m_0=\prod_{p/m} p^{k_p-1}\), if \(k_2<2\); \(m_0=\frac12\prod p^{k_p-1}\), if \(k_2\ge 2\). \(p_1p_2\ldots p_z=\prod_{p/m}p\). \(\delta_p\) is the exponent to which \(p\) belongs modulo \(\dfrac{m}{p^{k_p}}\). \(s_p\) is an arbitrary integer satisfying \(s_p\delta_p\ge k_p\). \(\Delta_p\) is the exponent to which \(p\ne 2\) belongs modulo \(\dfrac{m_0}{p^{k_p-1}}\). \(t_p\) is an arbitrary integer satisfying \(t_p\Delta_p\ge k_p-1\). The quantities \(\Delta_2\) and \(t_2\) are defined analogously. \(E^{(m)}\) is the group of residue classes modulo \(m\) relatively prime to \(m\). It is known that the group \(E^{(s)}\) for \(s=2^k,\ k\ge 2\), decomposes into the direct product of the cyclic group \(E_1^{(s)}\) of order two with generator \(a_1=-1\) and the cyclic group \(E_2^{(s)}\) of order \(2^{k-2}\). An arbitrary generator of the latter group in the case \(k=k_2=k_2(m)\ge 2\) will be denoted by \(a_2\). For example, \(a_2=5\). If, for \(\varepsilon\) relatively prime to \(m\), the condition
\[ \varepsilon=(-1)^{k\bmod 2}a_2^\delta(2^{k_2}) \]
is satisfied, we set \(\operatorname{ind}_2\varepsilon\) equal to the least of the possible (for the given \(a_2\)) natural \(\delta\)’s. \(a_p\) is a certain primitive root modulo \(p^{k_p}\), where \(p\) is an odd prime. \(\operatorname{ind}_p\varepsilon\) is the index of \(\varepsilon\) modulo \(p^{k_p}\) with base \(a_p\).
\[ \log x=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}(x-1)^k; \]
\[ e^x=\sum_{k=0}^{\infty}\frac{1}{k!}x^k \]
are formal power series.

We shall say that a series of this type converges modulo \(m\) at the point \(x=x_0\) if all its terms, starting from some one, are divisible by \(m\) at \(x=x_0\). Divisibility here is understood in the sense of divisibility of a rational fraction by a modulus. All the series occurring in the text converge modulo the corresponding moduli.

In this article \(\varepsilon\) everywhere denotes a number relatively prime to \(m\), and \(p\) denotes a prime and only a prime number.

1. Suppose first that \(m=p^k\), where \(p\) is an odd prime. For this case one can prove the formula
   \[ \operatorname{ind}_p\varepsilon\equiv \frac{\xi}{p}\log \varepsilon^{p-1} \equiv \frac{\xi}{p}\sum_{s=1}^{n}\frac{(-1)^{s-1}}{s}\left(\varepsilon^{p-1}-1\right)^s \equiv \]
   \[ \equiv \frac{\xi}{p}\sum_{s=1}^{n}\frac{(-1)^{s-1}}{s}C_n^s\left(\varepsilon^{(p-1)s}-1\right) \pmod {p^{k-1}}. \tag{1} \]
   Here \(n=n(k)\) is any natural number satisfying the condition
   \[ \frac{1}{s}p^{s-1}\equiv 0\pmod {p^{k-1}} \]
   for all \(s>n\).

\(\xi\) is relatively prime to \(p\), and

\[ \xi^{-1} \equiv \frac{1}{p}\log a_p^{p-1} \pmod {p^{k-1}}. \]

In particular, if \(\varepsilon^{p-1} \equiv 1 + pt \pmod {p^k}\), and \(a_p^{p-1} \equiv 1 + pt_0 \pmod {p^k}\), then

\[ \operatorname{ind}_p \varepsilon \equiv \xi\left(t-\frac{p}{2}t^2+\frac{p^2}{3}t^3-\frac{p^3}{4}t^4+\cdots\right)\pmod {p^{k-1}}, \]

where

\[ \xi^{-1}\equiv t_0-\frac{p}{2}t_0^2+\frac{p^2}{3}t_0^3-\frac{p^3}{4}t_0^4+\cdots\pmod {p^{k-1}}. \]

We note that one can choose the primitive root \(a_p\) so that the condition \(\xi \equiv 1 \pmod {p^{k-1}}\) is satisfied. For this it is enough to choose \(a_p\) (such a choice is always possible) satisfying the condition:

\[ a_p^{p-1}\equiv e^p\equiv 1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+\frac{p^4}{4!}+\cdots+\frac{p^s}{s!}+\cdots \pmod {p^k}. \]

II. Let us now consider the case \(m=2^k,\ k\ge 2\). It can be shown that

\[ 2+\frac{2^2}{2}+\frac{2^3}{3}+\frac{2^4}{4}+\frac{2^5}{5}+\cdots+\frac{2^n}{n}\equiv 0\pmod {2^k}, \tag{2} \]

starting from some \(n=n(k)\). Using this formula, the following facts are proved:

1. The projection \(E^{(m)}\) onto the cyclic direct factor \(E_2^{(m)}\) of order \(2^{k-2}\) is given by the formula:

\[ \pi(\varepsilon)\equiv e^{\log \varepsilon}\pmod m. \tag{3} \]

Here \(\log \varepsilon\) is taken modulo \(m\).

1. The formula holds

\[ \operatorname{ind}_2 \varepsilon \equiv \frac{\xi}{4}\log \varepsilon \equiv \frac{\xi}{4}\sum_{s=1}^{n}\frac{(-1)^{s-1}}{s}(\varepsilon-1)^s \equiv \]

\[ \equiv \frac{\xi}{4}\sum_{s=1}^{n}\frac{(-1)^{s-1}}{s}C_n^s(\varepsilon^s-1)\pmod {2^{k-2}}. \tag{4} \]

Here \(n=n(k)\) is any natural number satisfying the condition

\[ \frac{1}{s}2^{s-2}\equiv 0\pmod {2^{k-2}}\quad \text{for all } s>n. \]

\(\xi\) is relatively prime to \(2\), and

\[ \xi^{-1}\equiv \frac{1}{4}\log a_2\pmod {2^{k-2}}. \]

In particular, if

\[ \varepsilon\equiv 1+2t\pmod {2^k}, \quad a_2\equiv 1+2t_0\pmod {2^k}, \]

then

\[ \operatorname{ind}_2\varepsilon\equiv \xi\left(\frac{t(1-t_0)}{2}+\frac{2}{3}t^3-\frac{2^2}{4}t^4+\frac{2^3}{5}t^5-\cdots\right)\pmod {2^{k-2}}, \]

where

\[ \xi^{-1}\equiv \frac{t_0(1-t_0)}{2}+\frac{2}{3}t_0^3-\frac{2^2}{4}t_0^4+\frac{2^3}{5}t_0^5-\cdots \pmod {2^{k-2}}. \]

We note that, with \(a_2\) satisfying the condition

\[ a_2\equiv e^4\equiv 1+4+\frac{4^2}{2!}+\frac{4^3}{3!}+\frac{4^4}{4!}+\cdots \pmod {2^k}, \]

and only with such an \(a_2\), one has \(\xi\equiv 1 \pmod {2^{k-2}}\), and the formula for \(\operatorname{ind}_2\varepsilon\) takes its simplest form.

III. We now pass to the general case, where \(m\) is arbitrary. The well-known theorem asserting that

\[ E^{(m)}\cong \prod_{p/m} E(p^{k_p}) \]

gives us the decomposition

\[ E^{(m)}=E_1^{(m)}\times E_2^{(m)}\times \prod_{\substack{p\ne 2\\ p/m}} L_p^{(m)} \tag{5} \]

of the group \(E^{(m)}\) into the direct product of its cyclic subgroups:

\[ E_1^{(m)} \text{ is a cyclic group } \begin{cases} \text{trivial, if } k_2<2,\\ \text{of order }2,\text{ if } k_2\geqslant 2; \end{cases} \]

\[ E_2^{(m)} \text{ is a cyclic group of order } m_0; \]

\[ L_p^{(m)} \text{ is a cyclic group of order } p-1. \]

In the case \(m=2^k\) these notations coincide with those adopted earlier.
Let us first give a general characterization of the decomposition (5).

a) The subgroup \(E_1^{(m)}\) is generated by the class

\[ e_1 \equiv 2^{s_2\delta_2+1}-1 \pmod m. \]

b) \(E_2^{(m)}\) is the subgroup of \(E^{(m)}\) consisting of the classes \(\varepsilon \bmod m\) for which

\[ \varepsilon \equiv 1 \pmod {p_1p_2\cdots p_z}, \qquad \text{if } k_2<2, \]

\[ \varepsilon \equiv 1 \pmod {2p_1p_2\cdots p_z}, \qquad \text{if } k_2\geqslant 2. \]

c) The subgroup \(L_p^{(m)}\) is generated by the class \(e_p^{\,p^{k_p}-1}\), where

\[ e_p \equiv p^{s_p\delta_p}+\bigl(1-p^{s_p\delta_p}\bigr)a_p \pmod m. \]

d) \(E_3^{(m)}=E_1^{(m)}\times E_2^{(m)}\) is the subgroup of \(E^{(m)}\) consisting of the residue classes \(\varepsilon \bmod m\) for which \(\varepsilon \equiv 1 \pmod {p_1p_2\cdots p_z}\).

e) \(\prod\limits_{\substack{p\ne 2\\ p\mid m}} L_p^{(m)}\) is the subgroup of \(E^{(m)}\) consisting of the residue classes \(\varepsilon \bmod m\) for which \(\varepsilon^{p-1}\equiv 1 \pmod {p^{k_p}}\) for every odd prime \(p\) dividing \(m\).

f) The projection \(\pi_1:E^{(m)}\to E_3^{(m)}\) is given by the formula

\[ \pi_1(\varepsilon)\equiv \sum_{p\mid m}\bigl(1-p^{s_p\delta_p}\bigr)\varepsilon^{1-p^{k_p}-1} \equiv \prod_{p\mid m}\{p^{s_p\delta_p}+(1-p^{s_p\delta_p})\varepsilon\}^{1-p^{k_p}-1} \pmod m . \tag{6} \]

g) The projection \(\pi:E_3^{(m)}\to E_2^{(m)}\) is given by the formula:

\[ \pi(\varepsilon)\equiv \varepsilon^{\log\varepsilon}\pmod m . \tag{7} \]

IV. Let us now try to characterize the subgroup \(E_2^{(m)}\).

a) Let \(k_2<2\). An arbitrary element \(\varepsilon\) of the cyclic group \(E_2^{(m)}\) is represented in the form

\[ \varepsilon \equiv e^{p_1p_2\cdots p_z t} \equiv 1+p_1p_2\cdots p_z t+\frac{(p_1p_2\cdots p_z t)^2}{2!}+\cdots \equiv \bigl(e^{p_1p_2\cdots p_z}\bigr)^t \pmod m, \]

where \(t\) is uniquely determined modulo \(m_0\).

b) An arbitrary generator \(l\) of the cyclic group \(E_2^{(m)}\) in the case \(k_2<2\) is represented in any of the forms:

1) \[ l\equiv \prod_{p\mid m} e_p^{p-1}\pmod m,\qquad \text{where } e_p\equiv p^{s_p\delta_p}+\bigl(1-p^{s_p\delta_p}\bigr)a_p \pmod m. \]

2) \[ l\equiv 1+p_1p_2\cdots p_z t_0 \pmod m, \]

where \(t_0\) is relatively prime to \(m_0\) and is uniquely determined modulo \(m_0\).

3) \[ l\equiv e^{p_1p_2\cdots p_z t_1}\pmod m, \]

where \(t_1\) is relatively prime to \(m_0\) and is uniquely determined modulo \(m_0\).

Conversely, for any \(t_0\) and \(t_1\) relatively prime to \(m_0\), and any set \(\{a_p\}\) of primitive roots, formulas 1)—3) give certain generators of the group \(E_2^{(m)}\).

c) If \(k_2\geqslant 2\), the preceding assertions remain valid with \(p_1p_2\cdots p_z\) replaced by \(2p_1p_2\cdots p_z\).

In the following two propositions, \(l\equiv 1+p_1p_2\cdots p_z t_0^\delta \pmod m\) in the case \(k_2<2\), and \(l\equiv 1+2p_1p_2\cdots p_z t_0^\delta \pmod m\) in the case \(k_2\geqslant 2\), are certain generators of \(E_2^{(m)}\).

d) Let \(k_2<2\). Put \(1+p_1p_2\ldots p_z t \equiv (1+p_1p_2\ldots p_z t_0)\pmod m\). Then

\[ \delta \equiv \frac{\xi}{p_1p_2\ldots p_z}\log(1+p_1p_2\ldots p_z t)\equiv \]

\[ \equiv \xi\left(t-\frac{p_1p_2\ldots p_z}{2}t^2+\frac{(p_1p_2\ldots p_z)^2}{3}t^3-\ldots\right)\pmod {m_0}. \tag{8} \]

Here \((\xi,m_0)=1\) and

\[ \xi^{-1}\equiv \frac{1}{p_1p_2\ldots p_z}\log(1+p_1p_2\ldots p_z t_0)\pmod {m_0}. \]

An analogous assertion holds for the case \(k_2\ge 2\), with \(p_1p_2\ldots p_z\) replaced by \(2p_1p_2\ldots p_z\).

e) Let \(k_2<2\). Fix some system of primitive roots \(\{a_p\}\) and suppose that

\[ \prod_{p/m} e_p^{p-1}\equiv 1+p_1p_2\ldots p_z t_0\pmod m \]

for

\[ e_p\equiv p^{s_p\delta_p}+(1-p^{s_p\delta_p})a_p\pmod m. \]

If \(\varepsilon\equiv 1+p_1p_2\ldots p_z t\pmod m\), then

\[ \sum_{p/m_0}\frac{1-t_p^{\Delta_p}}{p-1}\operatorname{ind}_p\varepsilon \equiv \frac{\xi}{p_1p_2\ldots p_z}\log(1+p_1p_2\ldots p_z t)\pmod {m_0}, \tag{9} \]

where \((\xi,m_0)=1\) and

\[ \xi^{-1}\equiv \frac{1}{p_1p_2\ldots p_z}\log(1+p_1p_2\ldots p_z t_0)\pmod {m_0}. \]

An analogous assertion holds for the case \(k_2\ge 2\), with \(p_1p_2\ldots p_z\) replaced by \(2p_1p_2\ldots p_z\).

In addition to item e), note that the system \(\{a_p\}\) of primitive roots (we include \(a_2\) among them) can be chosen so that the condition \(\xi\equiv 1\pmod {m_0}\) is satisfied. For this it is enough to choose (which is always possible) \(\{a_p\}\) so that the relation

\[ \prod_{p/m} e_p^{p-1}\equiv \begin{cases} e^{p_1p_2\ldots p_z}\pmod m, & \text{if } k_2<2,\\ e^{2p_1p_2\ldots p_z}\pmod m, & \text{if } k_2\ge 2 \end{cases} \]

holds.

For the special case \(m=p^k\), item e) gives the well-known formula due to A. G. Postnikov,

\[ \frac{\operatorname{ind}(1+pt)}{p-1}\equiv \frac{\xi}{p}\log(1+pt)\pmod {p^{k-1}}, \tag{10} \]

as well as an explicit expression for \(\xi^{-1}\) modulo \(p^{k-1}\):

\[ \xi^{-1}\equiv \frac{1}{p}\log(1+pt_0)\pmod {p^{k-1}}, \qquad \text{if } a_p^{p-1}\equiv 1+pt_0\pmod {p^k}. \]

We note that formula (10), by virtue of the relation

\[ (p-1)\log(1+pt)\equiv \log(1+pt)^{p-1}\pmod {p^{k-1}}, \]

is nothing other than formula (1) for \(\varepsilon\equiv 1+pt\pmod {p^k}\). In fact these formulas are equivalent.

All the facts set forth above were obtained by the author in considering the properties of the exponential and logarithmic functions in the polyadic domain. Relation (2) is not new, although it was obtained by the author independently.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
29 XI 1961