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

3, 4, 5, 7, 15

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

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

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)

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

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).

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-François Alcover, Sep 21 2011 *)

Prog

(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

Crossrefs

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

Keyword

fini,full,nonn

Author

Lekraj Beedassy, Apr 25 2001

Extensions

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

Status

approved