Prachar's shifted-prime divisor paper

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balanced_divisors: Proves concentration and an entropy lower count for primorial divisors, providing the elementary repair needed in the conditional argument.

divisor_comparison: Derives the source corollary comparing the usual divisor function with the shifted-prime count along infinitely many even integers.

equation_17: States the conditional uniform lower prime count used by the source, with the modulus range and dependence on epsilon explicit.

equation_23: Proves the uniform logarithmic bound for the two-variable sieve weight, including its square means, shifted intervals, and dyadic endpoint cases.

hilfssatz_1: Records the precise uniform prime-progression estimate quoted from Rodosski and Tatuzawa; the analytic proof is external.

hilfssatz_2: Records the uniform Brun or Selberg upper bound for two distinct primitive linear forms; the degenerate equal-form case is excluded.

lcm_pairs: Proves the source remark that only O(x log x) ordered odd-prime pairs have the least common multiple of their shifts at most x.

satz_1: The odd shifted-prime divisor count has summatory function x log-log x plus a constant times x, with error O(x/log x).

satz_2: Infinitely many n have more than n to the c over log-log-squared n odd shifted-prime divisors; the full original counting proof is reconstructed.

satz_3: Reconstructs the stated one-half log-two lower bound under GRH, with an explicit divisor-selection repair of the abbreviated source proof.

satz_4: Proves the O(x log-squared x) second moment through the exact prime-pair parametrization and the uniform weighted sieve sum.

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K. Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p−1 haben, Monatshefte für Mathematik 59 (1955), 91–97, DOI 10.1007/BF01302992 (Crossref record read). The first printed page records receipt on 19 October 1954.

Source and convention

The copy read for this card is the Göttingen Digitalisierungszentrum article download: one archive cover followed by the seven printed pages 91–97. The archive record and article PDF were accessed. All eight physical pages were rendered and visually read for the full reconstruction; no OCR was used. The file prints on its prepended archive terms sheet (p. 1) the digitizer's notice that the library "provides access to digitized documents strictly for noncommercial educational, research and private purposes" and that "Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library"; the article pages print no copyright line, every other right reserved.

The source defines

$$\delta(n)=\#\{p\text{ an odd prime}:p-1\mid n\}.$$

Its prime-counting convention is retained in every main statement. Adding $p=2$ increases this function by one; that does not change the large-value conclusions but changes the linear constant in its first moment. All logarithms are natural, and asymptotic statements involving iterated logarithms use sufficiently large arguments.

Main theorems and complete proof chain

The four numbered theorems have complete reconstructions at the explicit external-input boundaries below.

• Satz 1 gives $\sum_{n\le x}\delta(n)=x\log\log x+Bx+O(x/\log x)$. Its floor-counting proof uses the classical prime harmonic estimate.
• Satz 2 gives $\delta(n)>\exp(c\log n/(\log\log n)^2)$ for infinitely many $n$, for some absolute $c>0$. The full original route includes deletion of an exceptional prime, exact disjoint divisor classes, residue-block counts, averaging over multiples, and conversion from $x$ to $n$.
• Satz 3 assumes GRH for Dirichlet $L$-functions and gives $\delta(n)>2^{(1/2-\epsilon)\log n/\log\log n}$ infinitely often for every fixed $\epsilon>0$. The abbreviated printed proof has a scale loss; the reconstruction supplies an explicit concentrated divisor selection and averaging repair, with every added deduction proved.
• Satz 4 gives $\sum_{n\le x}\delta(n)^2=O(x(\log x)^2)$. Its complete proof separates the prime diagonal, parametrizes the remaining pairs, and proves the full weighted sieve sum including shifted square means and the final dyadic interval.

The materially distinct lesser deductions are also complete: the O(x log x) least-common-multiple pair bound from pp. 96–97, and the comparison of the divisor function with delta along even integers on p. 96. The latter imports the classical positive main term for the second moment of the usual divisor function.

External analytic inputs and source precision

Hilfssatz 1 is the Rodosski–Tatuzawa prime-progression theorem with one possible exceptional divisor, quoted rather than proved by Prachar. Equation (17) is the precise GRH-conditional progression estimate. The proof of Hilfssatz 2, the Brun or Selberg estimate for two distinct primitive linear forms, is explicitly omitted in the paper. These are exact external analytic inputs, not unfilled same-paper proof steps. The classical prime-number theorem, prime harmonic estimate, and divisor-square moment have the same external status.

The reconstruction records these source issues and their demonstrated repairs without claiming an author-issued erratum:

• The p. 94 sentence counting multiples up to $x^2$ prints $\lfloor x/k\rfloor$; its own next display correctly averages over $\lfloor x^2/k\rfloor$.
• The literal primorial replacement offered for Satz 3 loses a factor two in the final coefficient because it retains $n\le x^2$. A biased divisor selection with $m=k/d$ proves the printed theorem, using the printed $x^{1/2-\epsilon}$ range of (17) only for moduli up to $x^{19/60}$; an exact integer comparison verifies the margin.
• Hilfssatz 2 requires distinct coefficients. The case $a=b=1$ is a single prime condition and leaves $g(0)$ undefined. The diagonal is handled separately, and the sieve is only applied with $N\ge2$.
• The display following (23) drops the factor $x$ from (21). Restoring it gives exactly the claimed $O(x(\log x)^2)$ second moment.

The correction note on p. 97 attributes a stronger least-common-multiple pair estimate $O(x\log\log x)$ to Erdős without printing his proof. That stronger estimate and the note's further representation-count assertion referring to Nöbauer are historical context only here. They are not certified by the four theorem proofs, and Nöbauer's formula is not imported as a checked input. The introductory classical comparisons for the divisor function are likewise not additional proof reconstructions.

Relation to the problem and later bounds

The unconditional theorem gives infinitely many integers with more than $n^{c/(\log\log n)^2}$ divisors of the form $p-1$. Erdős's complete product argument turns this into a lower bound for the first coprime pair of power differences. Erdős's 1974 summary misprints the input with an additional exponential and lists the reference year as 1954; this original article fixes both points. The Satz 2 page reconstructs the paper's proof in full; the analytic theorems that proof invokes remain external.

The later Fan–Pollack theorem strengthens the unconditional exponent to $(0.6736\log2)\log n/\log\log n$ and, under GRH, gives coefficient $\log((1+\sqrt5)/2)$ along an infinite sequence. Fan–Pollack count all primes, while Prachar excludes $2$; subtracting one preserves these comparisons after an arbitrarily small fixed loss in an exponential coefficient. Those stronger bounds are separate sources, not replacements for Prachar's two original methods.

Bears on. Problem 820: Satz 2 is the input to Erdős's product argument, which turns it into $H(n)>\exp(n^{c/(\log\log n)^2})$ for infinitely many $n$ and some $c>0$. That exponent $c/(\log\log n)^2$ is smaller than the exponent $(c-\epsilon)/\log\log n$ the problem asks about. The paper gives no upper bound for $H(n)$ and does not address whether $H(n)=3$ infinitely often. No formalization or local Lean build is claimed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.