A Number Theory Problem by Muhammad Rasel Parvej

1 1 2 2 + 1 1 4 2 + 1 1 6 2 + 1 1 8 2 + + 1 1 100 0 2 \dfrac{1}{1-2^{2}}+\dfrac{1}{1-4^{2}}+\dfrac{1}{1-6^{2}}+\dfrac{1}{1-8^{2}}+\cdots+\dfrac{1}{1-1000^{2}}\

The value of the expression above can be expressed as a b , \dfrac{a}{b}, where a a and b b are coprime integers.

What is the number of ways (equivalently, the number of ordered pair of integers ( a , b ) (a,b) ) the value of the expression above can be expressed as a b , \dfrac{a}{b}, where a a and b b are coprime integers.

Hint : Is it at all needed to compute the sum?


The answer is 2.

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1 solution

Clearly, every term of the series is negative. So, the sum will be negative too.

Now, if the sum is equal to a b \dfrac{a}{b} , with a a and b b are integers, then exactly one of a a and b b has to be negative.

As a a and b b are coprime, we have only one option for ( a , b ) (|a|,|b|) , which yields 2 options for ( a , b ) (a,b) : either ( a , b ) (-|a|,|b|) or ( a , b ) (|a|,-|b|) .

For the record, the sum, (evaluated using telescoping series), is 500 1001 -\dfrac{500}{1001} .

Brian Charlesworth - 4 years, 6 months ago

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