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A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution. 12
3, 4, 5, 7, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

Consider the form z = (x^2 - y^2) for x = (a^k + b^k) and y = (a^k - b^k). Now set a and b as the two complex solutions to the quadratic equation d^2 - d + 2 = 0. Then a = ((1-i*sqrt(7))/2), b = ((1+i*sqrt(7))/2), and z = (a^k + b^k)^2 - (a^k - b^k)^2 = ((((1-i*sqrt(7))/2)^(k) + ((1+i*sqrt(7))/2)^(k))^2 - (((1-i*sqrt(7))/2)^(k) - ((1+i*sqrt(7))/2)^(k))^2) = 4*a^k*b^k = 4*2^k = 2^(k + 2) for all k in N. If and only if k is in {1, 2, 3, 5, 13} = a(n) - 2, then y^2 = (a^k - b^k)^2 = -7, and z is therefore a power of two following form x^2 - (-7) = x^2 + 7. x^2 = (A002249(a(n) - 2))^2 = A227078(n). 7 = (sqrt(7)*A107920(a(n) - 2))^2. - Raphie Frank, Jan 06 2014

REFERENCES

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.

LINKS

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)

T. Do, Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n

Gerry Myerson, Bibliography

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, Diophantine equation

EXAMPLE

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.

MATHEMATICA

ramaNagell[n_] := Reduce[x^2 + 7 == 2^n, x, Integers] =!= False; Select[ Range[100], ramaNagell] (* Jean-Fran├žois Alcover, Sep 21 2011 *)

PROG

(MAGMA) [n: n in [0..100] | IsSquare(2^n-7)]; // Vincenzo Librandi, Jan 07 2014

CROSSREFS

Cf. A038198.

Sequence in context: A101759 A089560 A239547 * A216433 A101761 A035359

Adjacent sequences:  A060725 A060726 A060727 * A060729 A060730 A060731

KEYWORD

fini,full,nonn

AUTHOR

Lekraj Beedassy, Apr 25 2001

EXTENSIONS

Added keyword "full", M. F. Hasler, Feb 23 2009

STATUS

approved

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Last modified July 29 14:20 EDT 2014. Contains 245039 sequences.