This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A180445 All positive solutions, x, for each of two Diophantine equations noted by Richard K. Guy. 5
 1, 2, 3, 6, 91 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 2*(x^2)*((x^2)-1) = 3*((y^2)-1) has only these five positive solutions. x*(x-1)/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 Ramanujan-Nagell triangular numbers; z = {0, 1, 2, 4, 12} = (A060728(n) - 3). - Raphie Frank, Jun 26 2013 LINKS 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, 697-712. [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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 00:41 EST 2019. Contains 329816 sequences. (Running on oeis4.)