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A060728
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Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.
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16
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OFFSET
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1,1
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COMMENTS
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See A038198 for corresponding x. - Lekraj Beedassy, Sep 07 2004
Also numbers such that 2^(n-3)-1 is in A000217, i.e., a triangular number. - M. F. Hasler, Feb 23 2009
With respect to M. F. Hasler's comment above, all terms 2^(n-3) - 1 are known as the Ramanujan-Nagell triangular numbers (A076046). - Raphie Frank, Mar 31 2013
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
J. Roberts, Lure of the Integers. pp. 90-91, MAA 1992.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts (2002): 96-98.
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LINKS
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Table of n, a(n) for n=1..5.
T. Skolem, S. Chowla and D. J. Lewis, The Diophantine Equation 2^(n+2)-7=x^2 and Related Problems. Proc. Amer. Math. Soc. 10 (1959) 663-669. [M. F. Hasler, Feb 23 2009]
Anonymous, Developing a general 2nd degree Diophantine Equation x^2 + p = 2^n
M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM: item 31: A Ramanujan Problem (R. Schroeppel)
Curtis Bright, Solving Ramanujan's Square Equation Computationally
Spencer De Chenne, The Ramanujan-Nagell Theorem: Understanding the Proof
T. Do, Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n
A. Engel, Problem-Solving Strategies. p. 126.
Gerry Myerson, Bibliography
T. Nagell, The Diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961), no. 2-3, 185-187.
S. Ramanujan, Journal of the Indian Mathematical Society, Question 464(v,120)
Eric Weisstein's World of Mathematics, Ramanujan's Square Equation
Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers
Wikipedia, Carmichael's Theorem
Wikipedia, Diophantine equation
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FORMULA
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a(n) = log_2(8*A076046(n) + 8) = log_2(A227078(n) + 7)
Empirically, a(n) = Fibonacci(c + 1) + 2 = ceiling[e^((c - 1)/2)] + 2 where {c} is the complete set of positive solutions to {n in N | 2 cos(2*Pi/n) is in Z}; c is in {1,2,3,4,6} (see A217290).
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EXAMPLE
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The fifth and ultimate solution to Ramanujan's equation is obtained for the 15th power of 2, so that we have x^2 + 7 = 2^15 yielding x = 181.
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MATHEMATICA
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ramaNagell[n_] := Reduce[x^2 + 7 == 2^n, x, Integers] =!= False; Select[ Range[100], ramaNagell] (* Jean-François Alcover, Sep 21 2011 *)
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PROG
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(Magma) [n: n in [0..100] | IsSquare(2^n-7)]; // Vincenzo Librandi, Jan 07 2014
(PARI) is(n)=issquare(2^n-7) \\ Anders Hellström, Dec 12 2015
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CROSSREFS
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Cf. A002249, A038198, A076046, A077020, A077021, A107920, A215795, A227078
Sequence in context: A089560 A248077 A239547 * A330712 A295988 A216433
Adjacent sequences: A060725 A060726 A060727 * A060729 A060730 A060731
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KEYWORD
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fini,full,nonn
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AUTHOR
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Lekraj Beedassy, Apr 25 2001
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EXTENSIONS
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Added keyword "full", M. F. Hasler, Feb 23 2009
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STATUS
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approved
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