potm .pdf

File information


Original filename: potm.pdf

This PDF 1.5 document has been generated by TeX / pdfTeX-1.40.17, and has been sent on pdf-archive.com on 24/08/2016 at 03:36, from IP address 73.185.x.x. The current document download page has been viewed 381 times.
File size: 94 KB (1 page).
Privacy: public file


Download original PDF file


potm.pdf (PDF, 94 KB)


Share on social networks



Link to this file download page



Document preview


2016 September POTM
Solution by Benjamin Thomas
Prove: a, b ∈ Z+ and ab + 1 | a2 + b2 =⇒

a2 +b2
ab+1

= k 2 for k ∈ Z

a2 + b2
= j for a, b, j ∈ Z before establishing
ab + 1
2
a contradiction that proves j = k for integer k.
Let (a, b) = (A, B) be a solution for arbitrary integer j and A ≥ B. We will
assume that j is not a perfect square. Clearly if one solution exists, then infinitely
many exist, so we will choose (A, B) to minimize A + B.
Algebraic manipulation yields the equation A2 − jAB + B 2 − j = 0. Without
loss of generality, we will hold B, j constant, make the substitution x = A and view
this as a polynomial in x.
We begin by looking at the equation

Lemma 1: If r1 , r2 are roots of the polynomial f (x) = x2 + px + q, then
p = −(r1 + r2 ) and q = r1 r2 .
Proof:
If r1 , r2 are roots of f (x) we can write f (x) = (x − r1 )(x − r2 ) = x2 − (r1 + r2 )x +
r1 r2 . We then equate this with f (x) = x2 + px + q to complete the proof.
We now have x2 − (jB)x + (B 2 − j) = 0. Since (A, B) satisfies the original
equation, x1 = A yields one solution. By Lemma 1 we obtain x2 = jB − A and
2
x2 = B A−j . Since j, A, B are all integers, the first equation shows x2 ∈ Z. Our
assumption that j is not a perfect square implies that x2 6= 0.
Putting everything together, we have A ≥ B =⇒ x2 < A =⇒ x2 + B < A + B
which contradicts A + B being a minimum. This implies that j must be a perfect
square (j = k 2 for k ∈ Z)

1


Document preview potm.pdf - page 1/1


Related documents


potm
math words
ee
polyproof
oddmonomialproof
matrices and fibonacci

Link to this page


Permanent link

Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by e-Mail, Messenger, Whatsapp, Line..

Short link

Use the short link to share your document on Twitter or by text message (SMS)

HTML Code

Copy the following HTML code to share your document on a Website or Blog

QR Code

QR Code link to PDF file potm.pdf