Diophantine m-tuples

Classical references (Diophantus, Fermat, Euler):

  1. Diophanti Alexandrini, Arithmeticorum Libri Sex, cum commentariis C. G. Bacheti et observationibus D. P. de Fermat, Tolouse 1670; Lib. IV, q. XXI, p. 161.

  2. Diophantus von Alexandria, Die Arithmetik und die Schrift über Polygonalzahlen, (G. Wertheim, ed.), Teubner, Leipzig, 1890, pp. 104-107, 141-145.

  3. Diophante d'Alexandrie, Les six livres arithmetiques et le livre des nombres polygones, (P. ver Eecke, ed.), De Brouwer et Cie, Bruges, 1926; Paris, 1959, pp. 101-104, 134-137.

  4. T. L. Heath, Diophantus of Alexandria. A Study in the History of Greek Algebra. With a supplement containing an account of Fermat's theorems and problems connected with Diophantine analysis and some solutions of Diophantine problems by Euler, (Cambridge, England, 1910), Powell's Bookstore, Chicago; Martino Publishing, Mansfield Center, 2003, pp. 162-164, 177-182, 201-202, 344-349.

  5. Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers, (I. G. Bashmakova, ed.), Nauka, Moscow, 1974 (in Russian), pp. 103-104, 128-129, 232.

  6. J. Christianidis, J. Oaks, The Arithmetica of Diophantus. A Complete Translation and Commentary, Routledge, London, 2023, pp. 447, 469-470, 681-682, 711-713.

  7. P. Fermat, Observations sur Diophante, Oeuvres de Fermat, Vol. 1 (P. Tannery, C. Henry, eds.), 1891, p. 303.

  8. P. Fermat, Précis des Oeuvres Mathématiques, Éditions Jacques Gabay, 1989, pp. 62-64, 78.

  9. L. Euler, Opuscula Analytica I, 1783, pp. 329-344.

  10. L. Euler, Elements of Algebra, Springer, New York, 1972, pp. 432-435.

  11. A. H. Bell, C. C. Cross, Amer. Math. Monthly 5 (1898), 301-302.

  12. A. H. Bell, C. C. Cross, Euler, Amer. Math. Monthly 6 (1899), 85-86.

  13. L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, 1966, pp. 513-520.

  14. R. D. Carmichael, Diophantine Analysis, Dover, New York, 1959, pp. 106-112.

1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine m-tuples
6. Connections with elliptic curves
7. Various generalizations
8. References

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