A336819 Odd values of D > 0 for which the generalized Ramanujan-Nagell equation x^2 + D = 2^m has two or more solutions in the positive integers.
7, 15, 23, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 68719476735
Offset: 1
Keywords
Examples
For these exceptional cases, the corresponding solutions are: D = 7, (x,m) = (1,3), (3,4), (5,5), (11,7), (181,15); D = 23, (x,m) = (3,5), (45,11); D = 2^k -1, k >= 4, (x,m) = (1,k), (2^(k-1) - 1, 2*(k-1)). For k = 4 and D = 15, then 1^2 + 15 = 2^4 = 16, and 7^2 + 15 = 2^6 = 64. Remark: for k = 2 and D = 3, the two possible solutions corresponding to 2^k-1 coincide with (1, 2).
References
- Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, 2004, D10.
Links
- Roger Apéry, Sur une équation Diophantienne, C. R. Acad. Sci. Paris Sér. A251 (1960), 1263-1264.
- Roger Apéry, Sur une équation Diophantienne, C. R. Acad. Sci. Paris Sér. A251 (1960), 1451-1452.
- Frits Beukers, On the generalized Ramanujan-Nagell equation, I, Acta arithmetica, XXXVIII, 1980-1981, page 389-410.
- Wikipedia, Ramanujan-Nagell equation.
- Index entries for linear recurrences with constant coefficients, signature (3,-2).
Formula
From Colin Barker, Aug 05 2020: (Start)
G.f.: x*(7 - 6*x - 8*x^2 - 8*x^3 + 16*x^4) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>5.
a(n) = 2^(1+n)-1 for n>3. (End)
The two formulas with a(n) are true, according to theorem 2 of Beukers' link. - Bernard Schott, Aug 07 2020
Comments