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State True or False in each of the following statements.
If $$a$$ and $$b$$ are two positive integers greater than 20172017 and $$b - a > 4034$$ then there exist at least one pair $$a$$, $$b$$ such that the product $$(a - 2017)(b + 2017)$$ is greater than the product $$(a + 2017)(b - 2017)$$
Correct Answer: False
Expanding, $$(a - 2017)(b + 2017) - (a + 2017)(b - 2017) = 4034a - 4034b = 4034(a - b)$$. Since $$b > a$$, this is always negative, so the first product is always smaller. No such pair exists and the statement is False.
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