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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.

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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 12, 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.

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Pythagorean triple - Wikipedia

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is ( 3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime ( that is, ... When searching for integer solutions, the equation a2 + b2 = c2 is a ...

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

By definition, any number $n$ is a factor of itself. So, the largest number that divides ... Let $a$ and $b$ be positive integers such that ... Why must $\frac{a^2+b^2}{ab+1}$ 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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