A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.

3, 4, 5, 7, 15

Offset: 1

Author

Lekraj Beedassy, Apr 25 2001

Keywords

• fini
• full
• nonn

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

Interestingly enough, all the solutions correspond to noncomposite x, i.e., x = 1 for the first term, and primes 3, 5, 11, 181 for the following terms. - M. F. Hasler, Mar 11 2024

Examples

			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.
		

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

• 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

Crossrefs

Cf. A002249, A038198, A076046, A077020, A077021, A107920, A215795, A227078

Programs

Magma

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

Mathematica

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

PARI

is(n)=issquare(2^n-7) \\ Anders Hellström, Dec 12 2015

Formula

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).

Extensions

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