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Statement

The paper's equation (1) is $4/n=1/x+1/y+1/z$ "in natural numbers $x,y,z$ for any integer $n>1$" (p. 212), repetition allowed. Rosati's condition, quoted (p. 212): for a prime $n>3$, (1) is solvable if and only if "(2) $n=4ab(cd-b)-c$ or (3) $cn+1=4ab(cd-b)$ where $a,b,c$, and $d$ are natural numbers." The first algorithm finds, for a modulus $M$, the $N$ with $(M,N)=1$ and $0<N<M$ such that for primes "$(*)$ $n\equiv N\pmod M$ the Erdös--Straus conjecture may happen to be untrue" (p. 213); for every other class coprime to $M$ it has represented the progression $Ml+N$ by one of its formulas (4)--(7), each a rewriting of (2) or (3), so every prime in such a class satisfies (2) or (3).

The six classes modulo $840$ (p. 213, quoted): "when $M=840$ we obtain the result of K. Yamomoto: $n\equiv1,121,169,289,361,529\pmod{840}$".

Congruence (8) and Table 1 (p. 213, quoted): "A stronger condition is of the form: (8) $n\equiv N_1\pmod{9240}$ where $N_1$ is taken from Table 1 (34 numbers in all)." Table 1, as printed, read by columns:

$$\begin{gathered} 1,\ 169,\ 289,\ 361,\ 529,\ 841,\ 961,\ 1369,\ 1681,\ 1849,\ 2041,\ 2209,\ 2521,\ 2641,\ 2689,\ 2809,\ 3361,\\ 3481,\ 3529,\ 3721,\ 4321,\ 4489,\ 5041,\ 5161,\ 5329,\ 5569,\ 6169,\ 6241,\ 6889,\ 7561,\ 7681,\ 7921,\ 8089,\ 8761. \end{gathered}$$

Congruence (9) and Table 2 (p. 214, quoted): "To accelerate the calculations by the second algorithm described in the following, we give a still stronger condition on the possible unsolvability of the Erdös--Straus problem, namely (9) $n\equiv N_2\pmod{120120}$ where $N_2$ takes all 198 values from Table 2." Table 2, as printed, read by columns:

$$\begin{gathered} 1,\ 289,\ 361,\ 529,\ 841,\ 961,\ 1369,\ 1681,\ 1849,\ 2209,\ 2521,\ 2809,\ 3361,\ 3481,\ 3721,\ 4489,\ 5041,\ 5329,\ 6241,\ 6889,\\ 7921,\ 8089,\ 8761,\ 9409,\ 9601,\ 10201,\ 10609,\ 10921,\ 11449,\ 11881,\ 12601,\ 12769,\ 13729,\ 14569,\ 15409,\ 16129,\ 17161,\ 18001,\ 18769,\ 18841,\\ 19009,\ 19321,\ 20329,\ 20521,\ 21121,\ 21961,\ 22201,\ 22801,\ 23521,\ 24049,\ 24649,\ 26041,\ 26569,\ 27889,\ 28081,\ 28681,\ 29761,\ 29929,\ 30241,\ 31201,\\ 31249,\ 32041,\ 32761,\ 33049,\ 33289,\ 34609,\ 35281,\ 36481,\ 37129,\ 37249,\ 37489,\ 37801,\ 38809,\ 39601,\ 39649,\ 40681,\ 44521,\ 44641,\ 45049,\ 46201,\\ 46489,\ 47161,\ 48049,\ 48409,\ 48889,\ 49009,\ 49729,\ 49921,\ 50521,\ 51529,\ 51769,\ 52441,\ 53089,\ 53881,\ 54289,\ 54961,\ 55441,\ 55969,\ 56281,\ 56809,\\ 57121,\ 57961,\ 58081,\ 58249,\ 58969,\ 59929,\ 61681,\ 63001,\ 63361,\ 65209,\ 65521,\ 65641,\ 66049,\ 66361,\ 66889,\ 67369,\ 69001,\ 69169,\ 70009,\ 70249,\\ 70849,\ 71569,\ 72361,\ 72601,\ 73441,\ 73921,\ 74281,\ 74881,\ 76129,\ 76561,\ 76729,\ 77281,\ 77401,\ 78961,\ 79081,\ 79249,\ 80089,\ 80161,\ 80809,\ 81481,\\ 82681,\ 83329,\ 83521,\ 84529,\ 84841,\ 85201,\ 85801,\ 85849,\ 86641,\ 87481,\ 87649,\ 88201,\ 88321,\ 88729,\ 90721,\ 90841,\ 91081,\ 92401,\ 92569,\ 92689,\\ 94249,\ 94441,\ 95209,\ 96121,\ 96721,\ 97969,\ 98569,\ 98641,\ 99961,\ 100489,\ 101929,\ 102001,\ 103009,\ 103321,\ 103489,\ 104329,\ 105121,\ 105169,\ 105361,\ 106129,\\ 106681,\ 109201,\ 109321,\ 109561,\ 109729,\ 110881,\ 111049,\ 111409,\ 111721,\ 111841,\ 112561,\ 113401,\ 113569,\ 113689,\ 114409,\ 117049,\ 117121,\ 118561. \end{gathered}$$

The paper's abstract (p. 212) restates the outcome as: for a prime $n\not\equiv N\pmod M$, equation (1) is solvable, with $198$ such $N$ for $M=120120$.

Source. D. G. Terzi, On a conjecture by Erdös-Straus, BIT 11 (1971), 212--216; Rosati's conditions on printed p. 212 (PDF p. 1 of the publisher's scan), the algorithm, the six classes, (8) and Table 1 on p. 213 (PDF p. 2), (9) and Table 2 on p. 214 (PDF p. 3), read on the page images (the text layer reads the tables' digits cleanly except 5041 in Table 1 and 21961 in Table 2, each read with a letter for a digit, and garbles the formulas). The artifact is identified in the source digest.

Read depth. Claims checked: the statements quoted above and the three tables were read clause by clause and digit by digit on the page images on 2026-09-22. The algorithm (one paragraph) was read for structure only: the paper asserts that (2) is equivalent to each of (4)--(6) and (3) to (7) and works one substitution; the equivalences and the algorithm's runs were not checked. Filing observations, not review verdicts (checked here): Table 1 has 34 distinct entries and Table 2 has 198; every entry of Table 1 reduces modulo $840$ to one of the six classes; the entries of Table 2 reduce modulo $9240$ to exactly the 34 entries of Table 1; every entry of Table 2 is coprime to $120120$. Nothing here is independently reviewed.

Proof pointer

Page 213. With $\alpha,\beta,l$ natural numbers and $\delta(r)$ a divisor of $r$, the substitution $\alpha=b$, $\beta=cd-b$, $l=a$ turns (2) into (4) $n=4\alpha\beta l-\delta(\alpha+\beta)$, since $c=(\alpha+\beta)/d$ divides $\alpha+\beta$; the paper lists (5) $n=4\alpha\beta l-4\alpha\delta(\alpha)-\beta$ and (6) $n=(4\alpha\beta-1)l-4\alpha\delta(\alpha)$ as further rewritings of (2), and (7) $n=4\alpha\beta l-\delta(4\alpha\beta^2+1)$ as a rewriting of (3). For the modulus $M$ it sets $m=\delta(M)/4$ when $4\mid\delta(M)$ and $m=(\delta(M)+1)/4$ when $4\mid\delta(M)+1$, and for every factorization $m=\alpha\cdot\beta$ and every $N$ coprime to $M$ tests whether the progression $Ml+N$ is represented by one of (4)--(7); the classes never represented are the output. No further argument is printed.

Dependencies

Rosati's necessary and sufficient condition (2)--(3) for primes $n>3$ (Boll. Un. Mat. Ital. (3) 9 (1954), the paper's [3]; not held), and, for the six classes modulo $840$, Yamamoto's 1965 paper (the paper's [5]; not held). The problem page records the same six classes from the site's commentary, from the 2025 verification report and from the discussion thread under Mordell's name; the survey of Bloom and Elsholtz prints the list with $49$ in place of $529$ (p. 239, in the text before its Theorem 1), which the problem page reads as a misprint.

Bears on

• Problem 242: the partial result the site's commentary attributes to Terzi, "all $n$ outside $198$ bad classes modulo $120120$"; for a prime coprime to $120120$ outside the $198$ classes, (2) or (3) holds and $4/n$ is a sum of three unit fractions, which the page's Formulation converts into three distinct terms. The page's list of Mordell's six classes modulo $840$ is printed here first-hand, credited to Yamamoto.