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Source. The introductory condition (1) on printed p. 375 (here (2)), obtained from the theorem and remark on pp. 376–377 and the cyclic case of the corollary on p. 378 (PDF pp. 1–3). This is a complete rewritten deduction from the source's geometric theorem, with the strict limiting step and the one-prime case explicit.

Statement

Suppose a finite family $\{a_j\pmod{m_j}:1\le j\le k\}$ covers $\mathbb Z$, where the moduli $m_j>1$ are odd and pairwise distinct. Write

$$N=\operatorname{lcm}(m_1,\ldots,m_k)=\prod_{i=1}^n p_i^{s_i}, \qquad s_i\ge1,$$

with distinct odd primes $p_i$. For

$$x_i=\frac{p_i^{s_i}-1}{(p_i-2)p_i^{s_i}+1},\qquad F(x)=\prod_i(1+x_i)-\sum_i x_i,$$

the finite-exponent necessary condition is

$$F(x)\ge2. \tag{1}$$

In particular,

$$\prod_{i=1}^n\frac{p_i-1}{p_i-2} -\sum_{i=1}^n\frac1{p_i-2}>2. \tag{2}$$

Proof

By the cyclic coset correspondence, these classes cover $\mathbb Z/N\mathbb Z$ and map to proper prime-adic boxes with distinct cardinalities $N/m_j$. They are among the product sets permitted by the geometric theorem. That theorem proves (1). Alternatively, apply the nilpotent-group corollary directly to the cyclic group.

If $n=1$, then $F(x)=1$, contradicting (1). Thus a cover has $n\ge2$. For nonnegative coordinates,

$$\frac{\partial F}{\partial x_i} =\prod_{j\ne i}(1+x_j)-1\ge0,$$

and the derivative is strictly positive when $n\ge2$ and all the coordinates are positive. Furthermore, for every $p\ge3$ and $s\ge1$,

$$\frac1{p-2}-\frac{p^s-1}{(p-2)p^s+1} =\frac{p-1}{(p-2)((p-2)p^s+1)}>0.$$

Replacing each $x_i$ by $1/(p_i-2)$ strictly increases $F$. Combining this with (1) gives (2).

Source precision and limits

The source's remark writes a strict comparison between $\psi=F-1$ and its limiting expression without separating $n=1$. In that case $F$ is identically one, so the comparison is an equality. The direct one-prime exclusion above repairs this harmless endpoint before using strict monotonicity. No source-issued erratum is asserted.

These are necessary conditions only. The five-prime consequences and the comparison with Selfridge's condition are proved separately. Part II obtains a stronger obstruction by saving specific pairwise intersections rather than merely applying a union bound.

Bears on. Problem 7, without settling the existence of unrestricted distinct odd covering systems.