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Statement

The paper's equation (1) is $4/n=1/x+1/y+1/z$ in natural numbers (p. 212), and (2) is Rosati's form $n=4ab(cd-b)-c$ with $a,b,c,d$ natural numbers (p. 212). The second algorithm (p. 215) works on the assumption that every prime $n$ admits natural numbers $a,b,c,d$ satisfying (2). For a large bound $K$, a prime $n$ in one of the classes (9), and each natural number $m$ with $4m\le K$, it forms $s=m-n+m[n/m]$, which the paper identifies as the least strictly positive residue of $-n$ modulo $m$; if $s$ divides $\delta(m)+\delta'(m)$ for some factorization $m=\delta(m)\delta'(m)$, the paper says a decomposition of $n$ is found and the conjecture holds for $n$. The verification statement, quoted (p. 215):

With the help of the second algorithm the correctness of the Erdös--Straus conjecture is now proved for all $n\le10^8$.

The parenthetical list that follows credits the ranges, printed as a two-column table: R. Oblat, $n<106129$; A. Rosati, $106129\le n<141649$; K. Yamomoto (so printed), $n\le10^7$; D. Terzi, $10^7<n\le10^8$. The paper then invokes a remark it attributes to Obláth (printed "Oblat"): proving the conjecture for every prime $n$ proves it for every $n>1$. Table 3 is introduced as "the solutions of the Erdös--Straus problem for all primes $n$ of one of the forms given in Table 2, in the interval $10^7<n<10^8$", and lists, with columns $n,a,b,c,d$:

From a row of Table 3 the paper recovers the solution by $x=ab(cd-b)$, $y=nda(cd-b)$, $z=nabd$. It reports that the largest $K$ its computation needed was $141320$, at $n=43950481$, and that the algorithm was programmed in alpha-language and run on a BESM-6.

Source. D. G. Terzi, On a conjecture by Erdös-Straus, BIT 11 (1971), 212--216; printed p. 215 (PDF p. 4 of the publisher's scan), read on the page image; the text layer reads the table's digits cleanly and garbles the displayed formulas. The artifact is identified in the source digest.

Read depth. Claims checked: the passage was read clause by clause and Table 3 digit by digit on the page image on 2026-09-22. This is an author's report of a completed computation with no theorem label: the second algorithm is stated in one paragraph and was not reconstructed, the computation was not rerun, and the paper prints no record of its run beyond Table 3 and the value of $K$. Filing observations, not review verdicts (checked here): the seven $n$ are primes and lie in the classes of Table 2; the rows for $10330321$, $17330329$, $21021001$, $99822529$ and $99949441$ satisfy (2), and the printed formula gives three distinct unit fractions summing to $4/n$ (the formula gives $4/n$ whenever (2) holds, since $1/x+1/y+1/z=(n+c)/(nab(cd-b))$); the row for $34954921$ satisfies (2) with $a=118091$ in place of the printed $1118091$, a one-digit misprint; the row for $43950481$ satisfies neither (2) nor (3) as printed, and no change of one printed entry repairs it (searched over $b\le12$, $c<3000$, $d<400$), but it satisfies (2) with its $c$ and $d$ exchanged ($c=39$, $d=453$), a transposition misprint, and $4b(cd-b)=141320$ for this row equals the largest $K$ the paper reports, which occurred at this $n$. There are $43485$ primes in $(10^7,10^8)$ in the $198$ classes of Table 2 (sieve), so the seven rows are not the list the introducing sentence describes, and the paper does not say how they were chosen. Nothing here is independently reviewed.

Proof pointer

Page 215, one paragraph, paraphrased above. For a prime $n$ in one of the $198$ classes of (9), the algorithm runs over natural numbers $m$ with $4m\le K$, forms the least positive residue $s$ of $-n$ modulo $m$, and stops when $s$ divides $\delta(m)+\delta'(m)$ for some factorization $m=\delta(m)\delta'(m)$; the paper says this yields a decomposition of $n$ in the form (2) and prints no derivation. Primes outside the $198$ classes are covered by the first algorithm (Table 2), and composite $n$ by Obláth's remark that a solution for a prime scales to its multiples.

Dependencies

Congruence (9) and Table 2 (p. 214), Rosati's form (2) (p. 212), the verifications credited to Obláth ($n<106129$), Rosati ($106129\le n<141649$) and Yamamoto ($n\le10^7$), none held, and the computer run reported on p. 215. The problem page's history of finite verifications is Table 1 of Elsholtz and Tao (p. 4), which lists Terzi's $10^8$ with the caveat that his set of checked primes appears incomplete; the printed Table 3 is that incompleteness.

Bears on

• Problem 242: the $10^8$ entry of the finite-verification history, stated first-hand as an author's report; the seven printed quadruples are the only witnesses the paper gives for $10^7<n\le10^8$, two of them misprinted, and the later verifications to $10^{18}$ on the problem page supersede the range. The paper's convention allows repeated denominators; the page's Formulation converts a representation into one with three distinct terms.