Let a and b be positive integers such that ab+1 divides a2+b2. Show that a2+b2ab+1 is the square of an integer. ?

Step 1Supposing thata2+b2ab+1=k thena2−kba+b2−k=0 solving for aa=kb±k2b2−4(b2−k)2all the b's obeying b2=k and the a's obeyinga=kb are solutions. So(bak111284327946416512525621636⋯⋯⋯)The question follows.

Question: "Let a and b be positive integers such..."

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

Jayson Hebert

Answered question

2022-02-05

Let a and b be positive integers such that

a b + 1

divides

a^{2} + b^{2}

. Show that

\frac{a^{2} + b^{2}}{a b + 1}

is the square of an integer. ?

Answer & Explanation

traciaul6y

Beginner2022-02-06Added 14 answers

Step 1
Supposing that

\frac{a^{2} + b^{2}}{a b + 1} = k

then

a^{2} - k b a + b^{2} - k = 0

solving for a

a = \frac{k b \pm \sqrt{k^{2} b^{2} - 4 (b^{2} - k)}}{2}

all the b's obeying

b^{2} = k

and the a's obeying

a = k b

are solutions. So

(\begin{matrix} b & a & k \\ 1 & 1 & 1 \\ 2 & 8 & 4 \\ 3 & 27 & 9 \\ 4 & 64 & 16 \\ 5 & 125 & 25 \\ 6 & 216 & 36 \\ \dots & \dots & \dots \end{matrix})

The question follows.

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