I\'m having a difficult time doing proofs, so any help would be great, thanks. 1

ID: 2961562 • Letter: I

Question

I'm having a difficult time doing proofs, so any help would be great, thanks.

1)Show that there are no positive rational number x and y so that x^4 (+ or -) 1=y^2

2) Show that if n is a congruent number and m is an integer, then (m^2)n is congruent number.

3) Show that there are no right triangles with rational side-lengths whose area is a perfect square or twice a perfect square.

4) Show that every integer n (greater than or equal to) 3 occurs as a leg of some pythagorean triple.

Hint: The cases n even and nodd should be done separately.

let n is a congruent number.

there exists a right angle triangle with rational lengths p,q,r whose area is n.

p^2+q^2 = r^2

(mp)^2+(mq)^2 =(mr)^2

triangle with sides mp,mq,mr is also right angle triagle with area = m^2*n

thus proved

let n is an odd integer => n = 2k+1 for integer k

(2k+1)^2 = 4k^2+4k+1 = [(2k^2+2k+1)+(2k^2+2k)][(2k^2+2k+1)-(2k^2+2k)] = (2k^2+2k+1)^2-(2k^2+2k)^2.

n is side of a right triangle with other sides (2k^2+k+1),(2k^2+2k)

let n is evenn => n = 2k for some integer k

(2k)^2 = 4k^2 = 2*(2k^2) = (k^2+1)^2 - (k^2-1)^2

n is side of a right triangle with other sides (k^2+1), (k^2-1)

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.