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Topic: Can N^2 + N + 2 Be A Power Of 2?
Replies: 24   Last Post: Sep 19, 2002 6:50 PM

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 Gerry Myerson Posts: 431 Registered: 12/6/04
Re: Can N^2 + N + 2 Be A Power Of 2?
Posted: Sep 18, 2002 7:52 PM

In article <amaqif\$eve\$1@lydian.ccrwest.org>,
dmoews@xraysgi.ims.uconn.edu (David Moews) wrote:

=> In article <am9cci\$e4a\$1@kalman.eng.wayne.edu>,
=> David T. Ashley <dtashley@esrg.org> wrote:

=> |For N a natural number, can N^2 + N + 2 ever be a power of 2?
=> |
=> From mathscinet:
=>
=> 17,1055d
=> Browkin, Georges; Schinzel, Andre
=> Sur les nombres de Mersenne qui sont triangulaires. (French)
=> C. R. Acad. Sci. Paris 242 (1956), 1780--1781.
=>
=> It is proved that the only solutions in positive integers of
=> 2^x-1=y(y+1)/2 are (1,1), (2,2), (4,5) and (12,90).

In case anyone misses the connection: 2^r = N^2 + N + 2 ->
2^r - 2 = N^2 + N -> 2^(r - 1) - 1 = (N^2 + N)/2 = N(N + 1)/2.

=> 22 #25
=> Skolem, Thoralf; Chowla, S.; Lewis, D. J.
=> The diophantine equation 2^{n+2}-7=x^2 and related problems.
=> Proc. Amer. Math. Soc. 10 1959 663--669.
=>
=> Ramanujan [Collected papers, Cambridge Univ. Press, 1927, p. 327]
=> conjectured that the above equality has no rational integral
=> solutions for n and x except for n=1, 2, 3, 5, 13. Using Skolem's
=> p-adic method [8te Skand. Mat. Kongr. Forh., Stockholm, 1934, pp.
=> 163--188] the authors prove this conjecture and some related results.
=> Consider the sequence a_n for which a_0 = a_1 = 1; a_n =
=> a_{n-1}-2a_{n-2} for n >= 2. Then a_{n-1}^2=1 exactly for those
=> values of n such that 2^{n+2}-7=x^2 has a solution. An integer
=> appears in the sequence a_n at most three times.

In case anyone misses the connection: 2^r = N^2 + N + 2 ->
2^(r + 2) = 4N^2 + 4N + 8 = (2N + 1)^2 + 7.
--
Gerry Myerson (gerry@mpce.mq.edi.ai) (i -> u for email)