cp's OEIS Frontend

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

%I A060728 #99 Feb 16 2025 08:32:44
%S A060728 3,4,5,7,15
%N A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.
%C A060728 See A038198 for corresponding x. - _Lekraj Beedassy_, Sep 07 2004
%C A060728 Also numbers such that 2^(n-3)-1 is in A000217, i.e., a triangular number. - _M. F. Hasler_, Feb 23 2009
%C A060728 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
%C A060728 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
%D A060728 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
%D A060728 J. Roberts, Lure of the Integers. pp. 90-91, MAA 1992.
%D A060728 Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts (2002): 96-98.
%H A060728 T. Skolem, S. Chowla and D. J. Lewis, <a href="http://www.jstor.org/stable/2033452">The Diophantine Equation 2^(n+2)-7=x^2 and Related Problems</a>. Proc. Amer. Math. Soc. 10 (1959) 663-669. [_M. F. Hasler_, Feb 23 2009]
%H A060728 Anonymous, <a href="http://www.biochem.okstate.edu/OAS/OJAS/thiendo.htm">Developing a general 2nd degree Diophantine Equation x^2 + p = 2^n</a>
%H A060728 M. Beeler, R. W. Gosper and R. Schroeppel, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item31">HAKMEM: item 31: A Ramanujan Problem (R. Schroeppel)</a>
%H A060728 Curtis Bright, <a href="https://cs.uwaterloo.ca/~cbright/reports/ramanujans-square-equation.pdf">Solving Ramanujan's Square Equation Computationally</a>
%H A060728 Spencer De Chenne, <a href="http://buzzard.ups.edu/courses/2013spring/projects/spencer-ant-ups-434-2013.pdf">The Ramanujan-Nagell Theorem: Understanding the Proof</a>
%H A060728 T. Do, <a href="http://ojas.ucok.edu/98/T98/THIENDO.HTM">Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n</a>
%H A060728 A. Engel, <a href="https://bayanbox.ir/view/5143526036407318041/problem-solving-strategies-math-cs.blog.ir.pdf">Problem-Solving Strategies</a>. p. 126.
%H A060728 Gerry Myerson, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/ramanujan-nagell">Bibliography</a>
%H A060728 T. Nagell, <a href="https://projecteuclid.org/euclid.afm/1485893356">The Diophantine equation x^2 + 7 = 2^n</a>, Ark. Mat. 4 (1961), no. 2-3, 185-187.
%H A060728 S. Ramanujan, Journal of the Indian Mathematical Society, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q464.htm">Question 464(v,120)</a>
%H A060728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujansSquareEquation.html">Ramanujan's Square Equation</a>
%H A060728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation2ndPowers.html">Diophantine Equation 2nd Powers</a>
%H A060728 Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael%27s_theorem">Carmichael's Theorem</a>
%H A060728 Wikipedia, <a href="http://en.wikipedia.org/wiki/Diophantine_equation">Diophantine equation</a>
%F A060728 a(n) = log_2(8*A076046(n) + 8) = log_2(A227078(n) + 7)
%F A060728 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).
%e A060728 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.
%t A060728 ramaNagell[n_] := Reduce[x^2 + 7 == 2^n, x, Integers] =!= False; Select[ Range[100], ramaNagell] (* _Jean-François Alcover_, Sep 21 2011 *)
%o A060728 (Magma) [n: n in [0..100] | IsSquare(2^n-7)]; // _Vincenzo Librandi_, Jan 07 2014
%o A060728 (PARI) is(n)=issquare(2^n-7) \\ _Anders Hellström_, Dec 12 2015
%Y A060728 Cf. A002249, A038198, A076046, A077020, A077021, A107920, A215795, A227078
%K A060728 fini,full,nonn
%O A060728 1,1
%A A060728 _Lekraj Beedassy_, Apr 25 2001
%E A060728 Added keyword "full", _M. F. Hasler_, Feb 23 2009