Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-19T15:40:59.516Z Has data issue: false hasContentIssue false

On the Equation ax − by = 1. II

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge

Extract

The following conjecture was apparently first enunciated by Catalan (3) in 1844 but has never been proved.

Type
Articles
Copyright
Copyright © Cambridge Philosophical Society 1960

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

*(1)Carmichael, R. D., Amer. Math. Mon. 18 (1909), 34.Google Scholar
(2)Cassels, J. W. S., On the equation a xb x = 1. Amer. J. Math. 75 (1953), 159–62.CrossRefGoogle Scholar
(3)Catalan, E., Note extraite d'une lettre adressée à l'éditeur. J. reine angew. Math. 27 (1844), 192.Google Scholar
(4)Dickson, L. E., History of the theory of numbers, vol. 2 (New York, 1919).Google Scholar
(5)Euler, L., Theorematum quorundam arithemeticorum demonstrationes, Comm. Acad. Sci. Petropolitanae, 10 (1738), 125–46.Google Scholar
Reprinted in Comm. Arithm. Coll. 1 (1849), 2434 and inGoogle Scholar
Comm. Arithm. Coll. I (1915), 3858 (Collected Works, vol. 2).Google Scholar
*(6)Gerono, G. C., Note sur la résolution en nombres entiers et positifs de l'équation x m = y n + 1. Nouv. Ann. Math. (2) 9 (1870), 469–71 and 10 (1871), 204–6.Google Scholar
(7)Gloden, A., Histoire du ‘problème de Catalan’. Actes du septième congrès international d'histoire des sciences (Jerusalem, 1953), pp. 316–9.Google Scholar
(8)Gloden, A., Sur un problè me de Catalan. Mathesis, 61 (1952),302–3.Google Scholar
(9)Hampel, M., On the solution in natural numbers of the equation x my n = 1. Ann. Polon. Math. 3 (1956), 14.CrossRefGoogle Scholar
(10)Herschfeld, A., The equation 2x − 3y = d. Bull. Amer. Math. Soc. 42 (1936), 231–4.CrossRefGoogle Scholar
(11)Hoffmann, J. E., Neues über Fermats zahlentheoretische Herausforderungen von 1657. Abh. preuss. Akad. Wiss. 9 (1943), 152.Google Scholar
*(12)Lebesgue, V. A., Sur l'impossibilité en nombres entiers de l'équation x m = y 2 + 1. Nouv. Ann. Math. 9 (1850), 178–81.Google Scholar
(13)LeVeque, W. J., On the equation a xb y = 1. Amer. J. Math. 74 (1952), 325–31.Google Scholar
*(14)Nagell, T., Nordisk Mat. Tidskr. 1 (1919).Google Scholar
*(15)Nagell, T., Des équations indéterminées x 2 + x + 1 = y n et x 2 + x + 1 = 3y n. Norsk mat. Foren. Skr. Serie I, 2 (1921), 1214.Google Scholar
*(16)Nagell, T., Sur l'impossibilité de l'équation indéterminée z v + 1 = y 2. Norsk mat. Foren. Skr. Serie I, 4 (1921), 110.Google Scholar
*(17)Nagell, T., Sur une équation diophantienne à deux indéterminées. Norsk Vid. Selsk. Forh, Trondheim, 7 (1934), 137–9.Google Scholar
(18)Obláth, R., Az x 2 − 1 szamokrol. Math. phys. Lapok, 47 (1940), 5877.Google Scholar
(19)Obláth, R., über die Zahl x 2 − 1. Mathematica B, Zutphen, 8 (1940), 161–72.Google Scholar
(20)Obláth, R., Sobre ecuaciones diofanticas imposbiles de la forma x m + 1 = y n. Rev. mat. hisp.-amer. (iv), 1 (1941), 122–40.Google Scholar
(21)Obláth, R., über die Gleichung x m + 1 = y n. Ann. Polon. Math. 1 (1954), 73–6.CrossRefGoogle Scholar
(22)Pillai, S. S., On a xb y = c. J. Indian Math. Soc. 2 (1936), 119–22, 215.Google Scholar
(23)Rotkiewicz, A., Sur l'équation . Ann. Polon. Math. 3 (1956), 78Google Scholar
(24)Schinzel, A., Sur l'équation . Ann. Polon. Math. 3 (1956), 56.CrossRefGoogle Scholar
*(25)Selberg, S., Nordisk Mat. Tidskr. 14 (1932).Google Scholar
(26)Cestari, R., Risoluzione della diofantea X vZ t = 1. G. Mat. 85 (=V ser. 5), (1957), 197208.Google Scholar