Terzi 1971 conjecture erdos straus

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table_2: The 198 residue classes modulo 120120 that Terzi's first algorithm leaves as the only ones in which a prime n might lack a representation of 4/n as a sum of three unit fractions, with the 34 classes modulo 9240 and the six classes modulo 840 it refines.

verification_p215: Terzi's statement that the Erdős–Straus conjecture holds for all n up to 10^8, crediting Obláth, Rosati and Yamamoto below 10^7 and his own second algorithm between 10^7 and 10^8, with the seven printed Rosati quadruples of Table 3.

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D. G. Terzi, On a conjecture by Erdös-Straus, Nordisk Tidskrift for Informationsbehandling (BIT) 11 (1971), 212--216, DOI 10.1007/BF01934370; the running head prints "BIT 11 (1971), 212--216", the author's name also in Cyrillic, "Received September 23, 1970" (p. 212), and the author at the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (p. 216). Cited as [Te71] on the problem page. Its five references (p. 216) are Mordell, Diophantine equations (Academic Press, 1969), cited for the conjecture's formulation; the Russian translation of Davenport's The Higher Arithmetic (Moscow, 1965), cited for the connection with prime-representing polynomials and covering congruences; Rosati, Sull'equazione diofantea $4/n=1/x_1+1/x_2+1/x_3$, Boll. Un. Mat. Ital. (3) 9 (1954), no. 1; Bernstein, Zur Lösung der diophantischen Gleichung $m/n=1/x+1/y+1/z$, insbesondere im Fall $m=4$, J. reine angew. Math. 211 (1962), whose title the reference list prints with $m/n=l_1x+l_2y+l_3z$; and Yamamoto, On the diophantine equation $4/n=1/x+1/y+1/z$, Mem. Fac. Sci. Kyushu Univ. A19 (1965), no. 1. None of the five is held. The paper prints the second author of the conjecture as Straus and the last reference's author as "Yamomoto" throughout; the quotations below keep the printed spellings.

The copy read for this card is the publisher's scan of the printed article: 5 pages, printed pp. 212--216 = PDF pp. 1--5 (printed p. $n$ is PDF p. $n-211$), a 2005 scan (the file's metadata names a TIFF source and a June 2005 creation date) with an OCR text layer that locates the prose and reads the tables' digits cleanly except 5041 in Table 1 and 21961 in Table 2, each read with a letter for a digit, but garbles the Greek letters, subscripts, congruence signs and displayed formulas. Provenance: obtained from the publisher on 2026-09-22 as a DRM-free production PDF through the library's acquisition, from https://doi.org/10.1007/BF01934370; 200,366 bytes. No other version of the paper is known here. No notice is printed (pp. 212 and 216 read; the running head reads only "BIT 11 (1971), 212--216"); the publisher's article page shows "© BIT Foundations" under Rights and permissions (https://link.springer.com/article/10.1007/BF01934370, read 2026-10-02), and the Crossref record names only Springer's text-and-data-mining license, every other right reserved.

Read status: claims checked for the abstract, the formulation of the conjecture, Rosati's conditions (2) and (3) (p. 212), the target congruence $(*)$, equations (4)--(7), the definition of $m$, the six classes modulo $840$, congruence (8) and Table 1 (p. 213), congruence (9) and Table 2 (p. 214), the second algorithm, the verification statement with its attribution list, Table 3 and the solution formula (p. 215), each read clause by clause on the page images of PDF pp. 1--4 on 2026-09-22; the references and the address (p. 216) were read on the page image of PDF p. 5. The two algorithms are described in a paragraph each and were read for structure only: the equivalences (4)--(7) were not checked beyond the paper's own worked substitution, and the divisibility test of the second algorithm was not reconstructed. The three tables were checked arithmetically here as recorded below; the paper's computation was not rerun. Nothing here is independently reviewed.

Contents

• Abstract and introduction (p. 212, page image). The abstract announces two algorithms. The first, for a given modulus $M$, finds the residues $N$ such that (1) has a solution in natural numbers for every prime $n\not\equiv N\pmod M$; at $M=120120$ it leaves $198$ such $N$ (Table 2), which the abstract reads as the conjecture being "true with a probability greater than 0.99835". The second algorithm tests the conjecture for $10^7<n\le10^8$. The probability equals $1-198/120120$ to the printed precision (checked here) and is a heuristic remark, not a result. The conjecture is taken from Mordell's book as the solvability of "(1) $4/n=1/x+1/y+1/z$ in natural numbers $x,y,z$ for any integer $n>1$" (repetition allowed), with the main results credited to Rosati, Bernstein and Yamamoto; "This conjecture is still unproved." It is connected with prime-representing polynomials and covering congruences (Davenport, p. 57). Rosati's necessary and sufficient condition for a prime $n>3$, quoted: "(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 (pp. 212--213, page images). For a modulus $M$ it finds the $N$ with $(M,N)=1$, $0<N<M$ for which the paper cannot rule out a failure of the conjecture at primes $n\equiv N\pmod M$ (its congruence $(*)$). With $\alpha,\beta,l$ natural numbers and $\delta(r)$ a divisor of $r$, the paper states that (2) is equivalent to each of "(4) $n=4\alpha\beta l-\delta(\alpha+\beta)$, (5) $n=4\alpha\beta l-4\alpha\delta(\alpha)-\beta$, (6) $n=(4\alpha\beta-1)l-4\alpha\delta(\alpha)$" and (3) to "(7) $n=4\alpha\beta l-\delta(4\alpha\beta^2+1)$", the relations following "from (2) and (3) by suitable substitutions", of which one is worked: $\alpha=b$, $\beta=cd-b$, $l=a$ in (2) gives $n=4l\alpha\beta-c=4l\alpha\beta-(\beta+\alpha)/d=4l\alpha\beta-\delta(\alpha+\beta)$. For the fixed modulus $M$ the paper sets $m=\delta(M)/4$ if $4\mid\delta(M)$ and $m=(\delta(M)+1)/4$ if $4\mid\delta(M)+1$, and then, for each factorization $m=\alpha\beta$ and each $N$ coprime to $M$, tests whether some formula among (4)--(7) represents the progression $Ml+N$. Output: at $M=840$ the six classes $n\equiv1,121,169,289,361,529\pmod{840}$, which the paper credits to "K. Yamomoto"; a stronger condition (8), $n\equiv N_1\pmod{9240}$ with $N_1$ from Table 1 (34 numbers); and, to speed up the second algorithm, a still stronger condition (9), $n\equiv N_2\pmod{120120}$ with $N_2$ running over the $198$ values of Table 2 (p. 214, page image). Tables 1 and 2 are transcribed on the result page table_2. Filing observations, not review verdicts: 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, and every entry of Table 2 is coprime to $120120$ (checked here).
• The second algorithm and the verification (p. 215, page image). The algorithm assumes that every prime $n$ has natural numbers $a,b,c,d$ satisfying Rosati's (2) and searches for them by a divisor test: with a bound $K$ fixed and $n$ a prime in one of the classes (9), it runs through the natural numbers $m$ with $4m\le K$, sets $s=m-n+m[n/m]$, the least positive residue of $-n$ modulo $m$, and looks for an $m$ for which $s$ divides $\delta(m)+\delta'(m)$ for some factorization $m=\delta(m)\delta'(m)$; the paper says such an $m$ yields a decomposition of $n$ and hence the conjecture for $n$. The verification claim, quoted: "With the help of the second algorithm the correctness of the Erdös--Straus conjecture is now proved for all $n\le10^8$", followed by a parenthetical two-column attribution list: R. Oblat, $n<106129$; A. Rosati, $106129\le n<141649$; K. Yamomoto, $n\le10^7$; D. Terzi, $10^7<n\le10^8$. The restriction to prime $n$ rests on a remark the paper attributes to Obláth (printed "Oblat"), that the conjecture for primes implies it for every $n>1$. Table 3, 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 $10^7<n<10^8$", lists seven rows $(n,a,b,c,d)$: $(10330321,6980,5,79,1)$, $(17330329,131,2,447,37)$, $(21021001,24789,1,71,3)$, $(34954921,1118091,1,15,5)$, $(43950481,311,2,453,39)$, $(99822529,542514,1,47,1)$, $(99949441,265823,2,7,7)$, with the solution recovered by $x=ab(cd-b)$, $y=nda(cd-b)$, $z=nabd$. The paper reports that the largest $K$ its computation needed was $141320$, at $n=43950481$, and that the program was written in alpha-language and run on a BESM-6. Filing observations, not review verdicts: the seven $n$ are primes lying in the classes of Table 2; five rows satisfy (2) and the formula gives three distinct unit fractions summing to $4/n$; 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 here 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 $n$; and there are $43485$ primes in $(10^7,10^8)$ in the $198$ classes (sieve, checked here), so the seven rows are not the list the introducing sentence describes, and the paper does not say how they were chosen. The last point is the incompleteness that Table 1 of Elsholtz and Tao records against Terzi's $10^8$.
• References and address (p. 216, page image), listed above.

Compiled scope

The paper is compiled at statement depth for the two items Problem 242 consumes: the $198$ classes modulo $120120$ of congruence (9) and Table 2 (p. 214), paged on table_2 with Table 1 and the six classes modulo $840$, and the verification statement for $n\le10^8$ with Table 3 (p. 215), paged on verification_p215. The algorithms were read for structure only; the tables were checked arithmetically as recorded above; the computation was not rerun, and nothing here is independently reviewed.

Bears on. #242: congruence (9) with Table 2 (printed p. 214, PDF p. 3), "$n\equiv N_2\pmod{120120}$ where $N_2$ takes all 198 values from Table 2", is the partial result the site's commentary attributes to Terzi, "all $n$ outside $198$ bad classes modulo $120120$": for a prime $n$ coprime to $120120$ outside these classes the first algorithm has represented the class by one of (4)--(7), so Rosati's (2) or (3) holds and $4/n$ is a sum of three unit fractions; the paper allows repeated denominators, and the problem page's Formulation converts such a representation into one with three distinct terms. The same algorithm at $M=840$ prints the six classes $1,121,169,289,361,529$ (p. 213, PDF p. 2), the list the site gives for Mordell, here credited to Yamamoto. The verification statement (p. 215, PDF p. 4), "the correctness of the Erdös--Straus conjecture is now proved for all $n\le10^8$", is the $10^8$ entry of the finite-verification history, with the printed Table 3 covering seven of the primes it is said to cover, two of its rows misprinted. The paper proves neither the conjecture nor a counterexample and leaves the problem's status unchanged.

Results.

• Table 2 (p. 214, with (8) and Table 1 and the six classes modulo $840$ on p. 213): a prime $n$ can fail to have $4/n=1/x+1/y+1/z$ in natural numbers only if $n\equiv N_2\pmod{120120}$ for one of $198$ listed $N_2$.
• Verification, p. 215: the statement that the conjecture holds for all $n\le10^8$, the credit list, Table 3 and the solution formula; an author's report, not rerun.

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