4. Prove that every odd number is the difference of two perfect squares.
Java Write a program that will look for two perfect squares, both odd, that will add together to make another perfect square. Ex--->3^2+4^2=5^2 , 5^2+12^2=13^2 , 6^2+8^2=10^2 , 7^2+242^=25^2
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
Prove that in every simple graph there is a path from every vertex of odd degree to a vertex of odd degree.
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian. (ii) Deduce that the graph in Fig. 7.7 is non-Hamiltonian. Fig. 7.7 (iii) Show that, if n is odd, it is not possible for a knight to visit all the squares of an n chessboard exactly once by knight's moves and return to its starting point.
Describe the process of the difference of two squares. In your opinion what makes a difference of two squares easy to factor? First, define a perfect square. Then, indicate what to look for to confirm that you have a difference of squares. Finally , give an example, possibly one where you need to factor out a GCF ( greatest common factor) before using a difference of squares. How can we use difference of squares repeatedly to factor a difference of...
*Write a LISP program* Write a function that squares a number if it is odd and positive, doubles it if it is odd and negative, and otherwise divides the number by 2.
Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching
Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching
perfect sixth power. 9. Use the Fundamental Theorem of Arithmetic to prove that the product of any two odd integers is an odd integer.
8.20 Question. Which natural mumbers can be written as the sum of two squares of natural raumbers? State and prove the mast general theorem possible about which natural numbers can be written as the sum of two suares of nutural numbers, and prove it. We give the most gencral result next. 8.21 Theorem. A natural number n can be written as a sum of two squares of natural mumbers if and only if every prime congruent to 3 modulo 4...
Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1) (mod m) (m-1) (mod m [Hint : 1 i-(m-1 ) (mod m), 3 Ξ-(m-3) (mod ") , . .. , m-2 1-2 (mod m)] 14 (b) If p is an odd prime, prove that Hint: Use Part (a), and rearrange the Wilson's Theorem formula in two different ways