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The hundred years after Euler represented a period in which functions not satisfying his "official" constraints were frequently smuggled into mathematics through fudgy considerations involving infinite series expansions and the like. (Source: 1993, Steven J. Wagner, ‎Richard Wagner, Naturalism: A Critical Appraisal (page 72))

I tried to find out which of "the hundred years" and "the one hundred years" is more common, but Google search results mostly contain "the Hundred Years' War". "War" taken out, it seems both terms are in use. Do they mean the same thing? Does the sentence read the same with "the one hundred years"?

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Yes, they mean the same thing. In fact, you can do this with many lengths of time. "The thousand years", "The million years", etc, are all grammatically correct.

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