# Special Numbers

Standard

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, $n$, for which $x^n + y^n=z^n$ 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.

### 8 responses »

1. 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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2. and all Prime Numbers

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3. 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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