Erdős–Straus conjecture (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Erdős–Straus conjecture" in English language version.

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Graham (2013). Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions" (PDF), in Lovász, László; Ruzsa, Imre Z.; Sós, Vera T. (eds.), Erdös Centennial, Bolyai Society Mathematical Studies, vol. 25, Budapest: János Bolyai Mathematical Society, pp. 289–309, doi:10.1007/978-3-642-39286-3_9, MR 3203600
Obláth (1950); Elsholtz & Tao (2013) Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
See e.g. Sander (1994) for a simpler Diophantine formulation using more specific assumptions about which of $x$ , $y$ , and $z$ are divisible by $n$ . Sander, J. W. (1994), "On $4/n=1/x+1/y+1/z$ and Iwaniec' half-dimensional sieve", Journal of Number Theory, 46 (2): 123–136, doi:10.1006/jnth.1994.1008, MR 1269248.
Obláth (1950); Rosati (1954); Kiss (1959); Bernstein (1962); Yamamoto (1965); Terzi (1971); Jollensten (1976); Kotsireas (1999). Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Rosati, Luigi Antonio (1954), "Sull'equazione diofantea $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Boll. Un. Mat. Ital. (3) (in Italian), 9: 59–63, MR 0060526. Kiss, Ernest (1959), "Quelques remarques sur une équation diophantienne", Acad. R. P. Romîne Fil. Cluj Stud. Cerc. Mat. (in Romanian), 10: 59–62, MR 0125069. Bernstein, Leon (1962), "Zur Lösung der diophantischen Gleichung $m/n=1/x+1/y+1/z$ , insbesondere im Fall $m=4$ ", Journal für die Reine und Angewandte Mathematik (in German), 211: 1–10, doi:10.1515/crll.1962.211.1, MR 0142508, S2CID 118098315. Yamamoto, Koichi (1965), "On the Diophantine equation ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics, 19: 37–47, doi:10.2206/kyushumfs.19.37, MR 0177945. Terzi, D. G. (1971), "On a conjecture by Erdős-Straus", Nordisk Tidskr. Informationsbehandling (BIT), 11 (2): 212–216, doi:10.1007/BF01934370, MR 0297703, S2CID 124845157. Jollensten, Ralph W. (1976), "A note on the Egyptian problem", Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, vol. XVII, Winnipeg, Man.: Utilitas Math., pp. 351–364, MR 0429735. Kotsireas, Ilias (1999), "The Erdős-Straus conjecture on Egyptian fractions", Paul Erdős and his mathematics (Budapest, 1999), Budapest: János Bolyai Math. Soc., pp. 140–144, MR 1901903.
Bright & Loughran (2020). Bright, Martin; Loughran, Daniel (2020), "Brauer–Manin obstruction for Erdős–Straus surfaces", Bulletin of the London Mathematical Society, 52 (4): 746–761, arXiv:1908.02526, doi:10.1112/blms.12374, MR 4171399, S2CID 218959757.
Elsholtz & Tao (2013). Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Ionascu & Wilson (2011). Ionascu, Eugen J.; Wilson, Andrew (2011), "On the Erdös–Straus conjecture", Revue Roumaine de Mathématiques Pures et Appliquées, 56 (1): 21–30, arXiv:1001.1100, MR 2848047.
Webb (1970); Vaughan (1970); Li (1981); Yang (1982); Ahmadi & Bleicher (1998); Elsholtz (2001). Webb, William A. (1970), "On $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Proceedings of the American Mathematical Society, 25 (3), American Mathematical Society: 578–584, doi:10.2307/2036647, JSTOR 2036647, MR 0256984. Vaughan, R. C. (1970), "On a problem of Erdős, Straus and Schinzel", Mathematika, 17 (2): 193–198, doi:10.1112/S0025579300002886, MR 0289409 Li, Delang (1981), "On the equation $4/n=1/x+1/y+1/z$ ", Journal of Number Theory, 13 (4): 485–494, doi:10.1016/0022-314X(81)90039-1, MR 0642923. Yang, Xun Qian (1982), "A note on ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Proceedings of the American Mathematical Society, 85 (4): 496–498, doi:10.2307/2044050, JSTOR 2044050, MR 0660589. Elsholtz, Christian (2001), "Sums of $k$ unit fractions", Transactions of the American Mathematical Society, 353 (8): 3209–3227, doi:10.1090/S0002-9947-01-02782-9, MR 1828604.
Sierpiński (1956); Vaughan (1970). Sierpiński, Wacław (1956), "Sur les décompositions de nombres rationnels en fractions primaires", Mathesis (in French), 65: 16–32, MR 0078385. Reprinted with additional annotations in Sierpiński, Wacław (1974), Oeuvres Choisies, vol. I, Warsaw: PWN—Éditions Scientifiques de Pologne, pp. 169–184, MR 0414302. Vaughan, R. C. (1970), "On a problem of Erdős, Straus and Schinzel", Mathematika, 17 (2): 193–198, doi:10.1112/S0025579300002886, MR 0289409
Hofmeister & Stoll (1985). Hofmeister, Gerd; Stoll, Peter (1985), "Note on Egyptian fractions", Journal für die Reine und Angewandte Mathematik, 362: 141–145, MR 0809971.
Schinzel (1956); Suryanarayana & Rao (1965); Hagedorn (2000). Schinzel, André (1956), "Sur quelques propriétés des nombres $3/n$ et $4/n$ , où $n$ est un nombre impair", Mathesis (in French), 65: 219–222, MR 0080683. Suryanarayana, D.; Rao, N. Venkateswara (1965), "On a paper of André Schinzel", J. Indian Math. Soc., New Series, 29: 165–167, MR 0202659. Hagedorn, Thomas R. (2000), "A proof of a conjecture on Egyptian fractions", American Mathematical Monthly, 107 (1), Mathematical Association of America: 62–63, doi:10.2307/2589381, JSTOR 2589381, MR 1745572.

arxiv.org

Obláth (1950); Elsholtz & Tao (2013) Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Salez (2014). Salez, Serge E. (2014), The Erdős-Straus conjecture New modular equations and checking up to $N=10^{17}$ , arXiv:1406.6307, Bibcode:2014arXiv1406.6307S
Bright & Loughran (2020). Bright, Martin; Loughran, Daniel (2020), "Brauer–Manin obstruction for Erdős–Straus surfaces", Bulletin of the London Mathematical Society, 52 (4): 746–761, arXiv:1908.02526, doi:10.1112/blms.12374, MR 4171399, S2CID 218959757.
Elsholtz & Tao (2013). Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Ionascu & Wilson (2011). Ionascu, Eugen J.; Wilson, Andrew (2011), "On the Erdös–Straus conjecture", Revue Roumaine de Mathématiques Pures et Appliquées, 56 (1): 21–30, arXiv:1001.1100, MR 2848047.

doi.org

Graham (2013). Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions" (PDF), in Lovász, László; Ruzsa, Imre Z.; Sós, Vera T. (eds.), Erdös Centennial, Bolyai Society Mathematical Studies, vol. 25, Budapest: János Bolyai Mathematical Society, pp. 289–309, doi:10.1007/978-3-642-39286-3_9, MR 3203600
Obláth (1950); Elsholtz & Tao (2013) Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
See e.g. Sander (1994) for a simpler Diophantine formulation using more specific assumptions about which of $x$ , $y$ , and $z$ are divisible by $n$ . Sander, J. W. (1994), "On $4/n=1/x+1/y+1/z$ and Iwaniec' half-dimensional sieve", Journal of Number Theory, 46 (2): 123–136, doi:10.1006/jnth.1994.1008, MR 1269248.
Obláth (1950); Rosati (1954); Kiss (1959); Bernstein (1962); Yamamoto (1965); Terzi (1971); Jollensten (1976); Kotsireas (1999). Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Rosati, Luigi Antonio (1954), "Sull'equazione diofantea $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Boll. Un. Mat. Ital. (3) (in Italian), 9: 59–63, MR 0060526. Kiss, Ernest (1959), "Quelques remarques sur une équation diophantienne", Acad. R. P. Romîne Fil. Cluj Stud. Cerc. Mat. (in Romanian), 10: 59–62, MR 0125069. Bernstein, Leon (1962), "Zur Lösung der diophantischen Gleichung $m/n=1/x+1/y+1/z$ , insbesondere im Fall $m=4$ ", Journal für die Reine und Angewandte Mathematik (in German), 211: 1–10, doi:10.1515/crll.1962.211.1, MR 0142508, S2CID 118098315. Yamamoto, Koichi (1965), "On the Diophantine equation ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics, 19: 37–47, doi:10.2206/kyushumfs.19.37, MR 0177945. Terzi, D. G. (1971), "On a conjecture by Erdős-Straus", Nordisk Tidskr. Informationsbehandling (BIT), 11 (2): 212–216, doi:10.1007/BF01934370, MR 0297703, S2CID 124845157. Jollensten, Ralph W. (1976), "A note on the Egyptian problem", Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, vol. XVII, Winnipeg, Man.: Utilitas Math., pp. 351–364, MR 0429735. Kotsireas, Ilias (1999), "The Erdős-Straus conjecture on Egyptian fractions", Paul Erdős and his mathematics (Budapest, 1999), Budapest: János Bolyai Math. Soc., pp. 140–144, MR 1901903.
Bright & Loughran (2020). Bright, Martin; Loughran, Daniel (2020), "Brauer–Manin obstruction for Erdős–Straus surfaces", Bulletin of the London Mathematical Society, 52 (4): 746–761, arXiv:1908.02526, doi:10.1112/blms.12374, MR 4171399, S2CID 218959757.
Elsholtz & Tao (2013). Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Webb (1970); Vaughan (1970); Li (1981); Yang (1982); Ahmadi & Bleicher (1998); Elsholtz (2001). Webb, William A. (1970), "On $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Proceedings of the American Mathematical Society, 25 (3), American Mathematical Society: 578–584, doi:10.2307/2036647, JSTOR 2036647, MR 0256984. Vaughan, R. C. (1970), "On a problem of Erdős, Straus and Schinzel", Mathematika, 17 (2): 193–198, doi:10.1112/S0025579300002886, MR 0289409 Li, Delang (1981), "On the equation $4/n=1/x+1/y+1/z$ ", Journal of Number Theory, 13 (4): 485–494, doi:10.1016/0022-314X(81)90039-1, MR 0642923. Yang, Xun Qian (1982), "A note on ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Proceedings of the American Mathematical Society, 85 (4): 496–498, doi:10.2307/2044050, JSTOR 2044050, MR 0660589. Elsholtz, Christian (2001), "Sums of $k$ unit fractions", Transactions of the American Mathematical Society, 353 (8): 3209–3227, doi:10.1090/S0002-9947-01-02782-9, MR 1828604.
Sierpiński (1956); Vaughan (1970). Sierpiński, Wacław (1956), "Sur les décompositions de nombres rationnels en fractions primaires", Mathesis (in French), 65: 16–32, MR 0078385. Reprinted with additional annotations in Sierpiński, Wacław (1974), Oeuvres Choisies, vol. I, Warsaw: PWN—Éditions Scientifiques de Pologne, pp. 169–184, MR 0414302. Vaughan, R. C. (1970), "On a problem of Erdős, Straus and Schinzel", Mathematika, 17 (2): 193–198, doi:10.1112/S0025579300002886, MR 0289409
Schinzel (1956); Suryanarayana & Rao (1965); Hagedorn (2000). Schinzel, André (1956), "Sur quelques propriétés des nombres $3/n$ et $4/n$ , où $n$ est un nombre impair", Mathesis (in French), 65: 219–222, MR 0080683. Suryanarayana, D.; Rao, N. Venkateswara (1965), "On a paper of André Schinzel", J. Indian Math. Soc., New Series, 29: 165–167, MR 0202659. Hagedorn, Thomas R. (2000), "A proof of a conjecture on Egyptian fractions", American Mathematical Monthly, 107 (1), Mathematical Association of America: 62–63, doi:10.2307/2589381, JSTOR 2589381, MR 1745572.

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terrytao.files.wordpress.com

Obláth (1950); Elsholtz & Tao (2013) Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Elsholtz & Tao (2013). Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.

harvard.edu

ui.adsabs.harvard.edu

Salez (2014). Salez, Serge E. (2014), The Erdős-Straus conjecture New modular equations and checking up to $N=10^{17}$ , arXiv:1406.6307, Bibcode:2014arXiv1406.6307S

jstor.org

Webb (1970); Vaughan (1970); Li (1981); Yang (1982); Ahmadi & Bleicher (1998); Elsholtz (2001). Webb, William A. (1970), "On $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Proceedings of the American Mathematical Society, 25 (3), American Mathematical Society: 578–584, doi:10.2307/2036647, JSTOR 2036647, MR 0256984. Vaughan, R. C. (1970), "On a problem of Erdős, Straus and Schinzel", Mathematika, 17 (2): 193–198, doi:10.1112/S0025579300002886, MR 0289409 Li, Delang (1981), "On the equation $4/n=1/x+1/y+1/z$ ", Journal of Number Theory, 13 (4): 485–494, doi:10.1016/0022-314X(81)90039-1, MR 0642923. Yang, Xun Qian (1982), "A note on ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Proceedings of the American Mathematical Society, 85 (4): 496–498, doi:10.2307/2044050, JSTOR 2044050, MR 0660589. Elsholtz, Christian (2001), "Sums of $k$ unit fractions", Transactions of the American Mathematical Society, 353 (8): 3209–3227, doi:10.1090/S0002-9947-01-02782-9, MR 1828604.
Schinzel (1956); Suryanarayana & Rao (1965); Hagedorn (2000). Schinzel, André (1956), "Sur quelques propriétés des nombres $3/n$ et $4/n$ , où $n$ est un nombre impair", Mathesis (in French), 65: 219–222, MR 0080683. Suryanarayana, D.; Rao, N. Venkateswara (1965), "On a paper of André Schinzel", J. Indian Math. Soc., New Series, 29: 165–167, MR 0202659. Hagedorn, Thomas R. (2000), "A proof of a conjecture on Egyptian fractions", American Mathematical Monthly, 107 (1), Mathematical Association of America: 62–63, doi:10.2307/2589381, JSTOR 2589381, MR 1745572.

math.ca

cms.math.ca

Jaroma (2004). Jaroma, John H. (2004), "On expanding $4/n$ into three Egyptian fractions", Crux Mathematicorum, 30 (1): 36–37.

semanticscholar.org

api.semanticscholar.org

Obláth (1950); Elsholtz & Tao (2013) Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.
Obláth (1950); Rosati (1954); Kiss (1959); Bernstein (1962); Yamamoto (1965); Terzi (1971); Jollensten (1976); Kotsireas (1999). Obláth, Richard (1950), "Sur l'équation diophantienne ${\tfrac {4}{n}}={\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}$ ", Mathesis (in French), 59: 308–316, MR 0038999, M. Strauss [sic] a vérifié l'hypothèse de M. Erdős pour toute valeur de n < 5.000, et M. Shapiro pour n < 20.000. Nos théorèmes donnent la solution pour tout nombre < 106.128. Rosati, Luigi Antonio (1954), "Sull'equazione diofantea $4/n=1/x_{1}+1/x_{2}+1/x_{3}$ ", Boll. Un. Mat. Ital. (3) (in Italian), 9: 59–63, MR 0060526. Kiss, Ernest (1959), "Quelques remarques sur une équation diophantienne", Acad. R. P. Romîne Fil. Cluj Stud. Cerc. Mat. (in Romanian), 10: 59–62, MR 0125069. Bernstein, Leon (1962), "Zur Lösung der diophantischen Gleichung $m/n=1/x+1/y+1/z$ , insbesondere im Fall $m=4$ ", Journal für die Reine und Angewandte Mathematik (in German), 211: 1–10, doi:10.1515/crll.1962.211.1, MR 0142508, S2CID 118098315. Yamamoto, Koichi (1965), "On the Diophantine equation ${\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}$ ", Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics, 19: 37–47, doi:10.2206/kyushumfs.19.37, MR 0177945. Terzi, D. G. (1971), "On a conjecture by Erdős-Straus", Nordisk Tidskr. Informationsbehandling (BIT), 11 (2): 212–216, doi:10.1007/BF01934370, MR 0297703, S2CID 124845157. Jollensten, Ralph W. (1976), "A note on the Egyptian problem", Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, vol. XVII, Winnipeg, Man.: Utilitas Math., pp. 351–364, MR 0429735. Kotsireas, Ilias (1999), "The Erdős-Straus conjecture on Egyptian fractions", Paul Erdős and his mathematics (Budapest, 1999), Budapest: János Bolyai Math. Soc., pp. 140–144, MR 1901903.
Bright & Loughran (2020). Bright, Martin; Loughran, Daniel (2020), "Brauer–Manin obstruction for Erdős–Straus surfaces", Bulletin of the London Mathematical Society, 52 (4): 746–761, arXiv:1908.02526, doi:10.1112/blms.12374, MR 4171399, S2CID 218959757.
Elsholtz & Tao (2013). Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943.

terrytao.wordpress.com

On the number of solutions to 4/p = 1/n_1 + 1/n_2 + 1/n_3, Terence Tao, "What's new", July 7, 2011; Counting the number of solutions to the Erdös-Straus equation on unit fractions, Terence Tao, July 31, 2011.

uci.edu

ics.uci.edu

Eppstein (1995), conflict resolution section. Eppstein, David (1995), "Ten algorithms for Egyptian fractions", Mathematica in Education and Research, 4 (2): 5–15. See in particular the "Small numerators" section
Eppstein (1995). Eppstein, David (1995), "Ten algorithms for Egyptian fractions", Mathematica in Education and Research, 4 (2): 5–15. See in particular the "Small numerators" section
See the conflict resolution section of Eppstein (1995) for a proof that a closely related replacement process (with a different expansion for even denominators that reduces the number of fractions) always terminates with a non-repeating expansion. Eppstein, David (1995), "Ten algorithms for Egyptian fractions", Mathematica in Education and Research, 4 (2): 5–15. See in particular the "Small numerators" section

ucsd.edu

math.ucsd.edu

Graham (2013). Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions" (PDF), in Lovász, László; Ruzsa, Imre Z.; Sós, Vera T. (eds.), Erdös Centennial, Bolyai Society Mathematical Studies, vol. 25, Budapest: János Bolyai Mathematical Society, pp. 289–309, doi:10.1007/978-3-642-39286-3_9, MR 3203600