sometimes called the Ramanujan-Nagell equation, has any solutions other than ,
4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; OEIS A060728).
These correspond to , 3, 5, 11, and 181 (OEIS A038198).
Nagell (1948) and Skolem et al. (1959) showed there are no solutions past
,
thus establishing Ramanujan's question in the negative.
A generalization to two variables and was considered by Euler (Engel 1998, p. 126).
Bundschuh, P. "On the Diophantine Equation of Ramanujan-Nagell." In Seminar on Diophantine Approximation. Papers from the Seminar Held in Yokohama,
April 6-8, 1987. Yokohama, Japan: Keio University, Department of Mathematics,
pp. 31-40, 1988.Cohen, E. L. "On the Ramanujan-Nagell
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."
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L. J. Diophantine
Equations. New York: Academic Press, p. 205, 1969.Nagell,
T. Nordisk Mat. Tidskr.30, 62-64, 1948.Nagell, T "The
Diophantine Equation ." Arkiv för Mat.4, 185-187,
1960.Ramanujan, S. Collected
Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 327, 2000.Ramasmay,
A. M. S. "Ramanujan's Equation." J. Ramanujan Math. Soc.7,
133-153, 1992.Roberts, J. The
Lure of the Integers. Washington, DC: Math. Assoc. Amer., pp. 90-91,
1992.Schroeppel, R. C. Item 31 in Beeler, M.; Gosper, R. W.;
and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, p. 14, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item31.Skolem,
T.; Chowla, S.; and Lewis, D. J. "The Diophantine Equation and Related Problems." Proc. Amer. Math.
Soc.10, 663-669, 1959.Sloane, N. J. A. Sequences
A038198 and A060728
in "The On-Line Encyclopedia of Integer Sequences."Stewart,
I. and Tall, D. Algebraic
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"A Note on the Ramanujan-Nagell Equation, in Number-Theoretic Analysis."
In Number-Theoretic
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Technical University of Vienna, Vienna, 1988-1989 (Ed. H. Hlawka and
R. F. Tichy). Berlin: Springer-Verlag, pp. 206-207, 1990.
,
4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; OEIS A060728).
These correspond to 
, 3, 5, 11, and 181 (OEIS A038198).
Nagell (1948) and Skolem et al. (1959) showed there are no solutions past
,
thus establishing Ramanujan's question in the negative.
and 
was considered by Euler (Engel 1998, p. 126).
." Amer. Math. Monthly 94. 59-62,
1987.Mignotte, M. "Une nouvelle résolution de l'équation
."
Rend. Sem. Fac. Sci. Univ. Cagliari 54, 41-43, 1984.Mordell,
L. J. 
." Arkiv för Mat. 4, 185-187,
1960.Ramanujan, S. 
and Related Problems." Proc. Amer. Math.
Soc. 10, 663-669, 1959.Sloane, N. J. A. Sequences