(Solved) Does a positive integer exist with infinite factors?

tufalafo

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10 years ago

Does a positive integer exist with infinite factors?

I am teaching an introductory course in Computer Science. We defined an algorithm as terminating and asked if finding all factors of a positive integer was an algorithm. One student insisted that infinitely large integers exist that have an infinite number of factors. Do such numbers exist? Where can I find more info?

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leeta

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10 years ago

No. All factors must be smaller (or equal) to the integer you're factoring. Thus, no finite number can have an infinite number of factors.

However, you can name any arbitrarily large number, and come up with an integer that has that many factors. That might make it seem like you could come up with an integer with infinitely many factors, but you can't. Infinity is not an integer.

Hope that helps!

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tryingtopass

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10 years ago

it does not

a number N can have at most N factors

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fisher

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10 years ago

No matter how large an integer is, it is still finite and can only have a finite number of factors. You can google finite if you are interested.

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dashaalynn

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10 years ago

This verified answer contains over 170 words.

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LIGHT

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10 years ago

There is no such thing as an "infinitely large integer."

There are infinitely large numbers. Some are cardinals, and some are ordinals, but none are integers.

My favorite introduction to infinitely large numbers is George Gamow's "One, two, three ... infinity," which is accessible and appealing. For mathematical rigor you can find a discussion of "transfinite numbers" in just about any textbook of elementary set theory.

And of course there are Wikipedia articles, which I've referenced below.

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