#### Let a,b be two positve integers such that ab+1 divides a^{2}+b^{2}. Show that \frac{a^{2}+b^{2}}{ab+1} is the square of an integer.

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**Let** a and **b** be **positive integers such** that **a b** + **1 divides** a **2** + **b**

Alon Amit, PhD in **Mathematics**; Mathcircler. ... on forums **such** as
artofproblemsolving, **math**.stackexchange and elsewhere. .... If we **have one**
solution of a quadratic equation, we **have** to be curious about the other. **Let's** ......
**1** is a perfect **square**. ...... Then, when **1**+**ab** is a divisor of a^**2**+**b**^**2**, there must be
a **positive integer** N ...

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Alternative proof that ( a **2** + **b 2** ) / ( **a b** + **1** ) is a **square** when it's an

**Mathematics** .... This is, IMHO, **one** of the most popular (and actually the most
beautiful) ... You can find it in these links (most of the solutions are the same as
the **two** you .... that a , **b** are **integers** (WOLOG we **have** chosen as **positive**
**integers**) and hence if c is ... **So** we **have** ... Now in order to make c as **integer**, we
must **have**.

For more information, see Alternative proof that ( a **2** + **b 2** ) / ( **a b** + **1** ) is a **square** when it's an

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**Square** root **of 2** - Wikipedia

The **square** root **of 2**, or the (**1**/**2**)th power **of 2**, written in **mathematics** as √**2** or **2**
^{1⁄2}, is the **positive** .... Another early close approximation is given in ancient
Indian **mathematical** .... The value of **b** cannot be **1** as there is no **integer** a the
**square** of which is **2**. ... **Let b** be the least **positive integer** for which √**2** is a
rational a/**b**.

For more information, see **Square** root **of 2** - Wikipedia

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Pythagorean triple - Wikipedia

A Pythagorean triple consists of three **positive integers** a, **b**, and c, **such** that a^{2} +
**b ^{2}** = c

For more information, see Pythagorean triple - Wikipedia

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What is the greatest **integer** that **divides** 101^100 - **1**? [Must be from ...

By definition, any number [**math**]n[/**math**] is a factor of itself. So, the largest
number that **divides** ... **Let** [**math**]a[/**math**] and [**math**]**b**[/**math**] be **positive integers**
**such** that ... Why must [**math]\frac**{a^**2**+**b**^**2**}{**ab**+**1}[/math**] be a perfect **square**? ...
which leave a remainder **1** when **divided** by 3 and a remainder **of 2** when **divided**
by 4?

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**Let** a and **b** be **positive integers such** that **a b** + **1 divides** a **2** + **b**

Hi I **have** been working on this problem for a while and don't understand the
solution given. ... **one** or more solution to the given condition where k is not a
perfect **square**. For a given value of k, **let** (A,**B**) be the solution to this equation
that minimizes .... came to the left side of the equality means that the other root is
an **integer**.

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