Recently I realized the special properties of following numbers:

**Zero (0)**

It is the only number which is both real and purely imaginary at same time.

**One (1)**

It is sufficient to create all the counting numbers (a.k.a. natural numbers).

**Two (2)**

This is the maximum exponent, , for which has solution in natural numbers. This peculiar property leads to “Fermat’s Last Theorem”.

*If you also have some special numbers in mind, please do share them below as comments.*

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All natural numbers are special. Don’t believe me? OK, assume that there exists a nonempty set of non-special natural numbers. Then there must be a lowest non-special number, which seems like a pretty special quality to me! This is a contradiction, therefore our assumption is false and the set of non-special numbers must be empty. QED

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I really liked your argument 🙂

Here definition of “special numbers” is ambiguous. One can’t prove/disprove any argument about subjective property (“…pretty special quality to me!…”). See: https://gaurish4math.wordpress.com/2015/11/05/metamathematics/

Your idea of proof is similar to Cantor’s Theorem stating “there can’t exist any surjection between a set and its power set”.

Glad to know that for you, all natural numbers are special. 🙂

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Today, luckily, I found this argument being discussed as a fallacy in elementary number theory by Martin Gardner on pp. 148 of his book “Hexaflexagons and Other Mathematical Diversions”.

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and all Prime Numbers

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Perfect numbers:

———————

6=1+2+3

28=1+2+4+7+14

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I wish to see “individual number” with unique properties. It’s true that 6 is a perfect number, but will it make any difference to existence of other perfect numbers (if 6 didn’t exist)? For examples, if 1 doesn’t exist we can’t generate all natural numbers.

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