Archimedes Laboratory Project™
Puzzles & Mental Activities that Enhance Critical and Creative Thinking Skills

› Home › Puzzle Archive › The Author

	Diophantine Cubic. Solve the equation below in integers! (Find at least 3 distinct roots): x3 + 1 = y2 Résolvez l’équation en nombres entiers: x3 + 1 = y2 Risolvere l’equazione in numeri interi: x3 + 1 = y2 You can comment and discuss this puzzle on our FaceBook or Blogger web page. › Click to show/hide the solution Obviously, one solution is (x, y) = (-1, 0). x cannot have any other negative value since y2 is positive. y2 = x3 + 1 = (x + 1)(x2 – x + 1) = = (x + 1)2 · (x2 – x + 1)/(x + 1) To solve the problem we must defactorize and analyze the last set of binomials: (x2– x + 1)/(x + 1) = = (x2 + x – 2x + 1)/(x + 1) = = x – [(2x + 1)/(x + 1)] = = x – [(2x - 2 + 3)/(x + 1)] = = x – 2 + 3/(x + 1) Now, 0 and 2 are the only integers for which 3/(x + 1) is an integer. So, the only solutions for (x, y) other than (-1, 0) are (0, +/-1) and (2, +/-3). Please spread the word about the Sunday Puzzle!

---

Text and images by Gianni A. Sarcone 
Labels: SUNDAY PUZZLE, puzzles to solve, recreational mathematics, weight
© G. Sarcone, Archimedes' Lab | Privacy & Terms |