

A180445


All positive solutions, x, for each of two Diophantine equations noted by Richard K. Guy.


4




OFFSET

1,2


COMMENTS

2*(x^2)*((x^2)1) = 3*((y^2)1) has only these five positive solutions.
x*(x1)/2 = (2^z)1 has only these five positive solutions.
Richard K. Guy notes, as Example 29: "True, but why the coincidence?"
Algebraically, y solutions = {1, 3, 7, 29, 6761} can be derived from x solutions as follows: y = sqrt(((2*x^2  1)^2 + 5)/6). From this relationship it becomes clear that the form (((2*x^2  1)^2 + 5)/6) can only be an integer square for x is in {1, 2, 3, 6, 91}. Thus, x and y solutions are also unique integer solutions to the following equivalency: (2x^2  1)^2 = 6y^2  5. From this relationship the following statement naturally follows: ((sqrt(6*y^2  5) + 1)/2  sqrt((sqrt(6*(y^2)  5) + 1)/2))/2 = (2^z  1) = {0, 1, 3, 15, 4095} = A076046(n), the RamanujanNagell triangular numbers; z = {0, 1, 2, 4, 12} = (A060728(n)  3).  Raphie Frank, Jun 26 2013


LINKS

Table of n, a(n) for n=1..5.
R. K. Guy, editor, Western Number Theory Problems, 19851221 & 23, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission. See Problem 85:08.
Richard K. Guy, The Strong Law of Small Numbers (example #29).
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]


FORMULA

x = sqrt((sqrt(6*(y^2)  5) + 1)/2) = (sqrt(2^(z + 3)  7) + 1)/2; y = {1, 3, 7, 29, 6761} and z = (A060728(n)  3) = A215795(n) = {0, 1, 2, 4, 12}.  Raphie Frank, Jun 23 2013


CROSSREFS

Cf. A076046, A060728.
Sequence in context: A018474 A018488 A018506 * A018531 A018546 A018553
Adjacent sequences: A180442 A180443 A180444 * A180446 A180447 A180448


KEYWORD

fini,full,nonn


AUTHOR

Jonathan Vos Post, Sep 05 2010


STATUS

approved



